Geometric measure theory and real analysis 9788876425226, 9788876425233

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5 0 3 1284 6

Geometric Measure

7

Theory and Real Analysis

9

edited by Luigi Ambrosio

EDIZIONI DELLA NORMALE

17

CRM SERIES

Vladimir I. Bogachev Department of Mechanics and Mathematics, Moscow State University, 119991 Moscow, Russia and St.-Tikhon’s Orthodox Humanitarian University Moscow, Russia Roberto Monti Dipartimento di Matematica Universit`a di Padova Via Trieste, 63 35121 Padova, Italia Emanuele Spadaro Max-Planck-Institut f¨ur Mathematik in den Naturwissenschaften Inselstrasse 22 D-04103 Leipzig Davide Vittone Dipartimento di Matematica Universit`a di Padova Via Trieste, 63 35121 Padova, Italia

Geometric Measure Theory and Real Analysis edited by Luigi Ambrosio

c 2014 Scuola Normale Superiore Pisa  ISBN 978-88-7642-522-6 ISBN 978-88-7642-523-3 (eBook)

Contents

Preface

ix

Vladimir I. Bogachev Sobolev classes on infinite-dimensional spaces

1 Measures on inVnite-dimensional spaces . . . . . . 2 Gaussian measures . . . . . . . . . . . . . . . . . 3 Integration by parts and differentiable measures . . 4 Sobolev classes over Gaussian measures . . . . . . 5 Inequalities and embeddings . . . . . . . . . . . . 6 Sobolev classes over differentiable measures . . . . 7 The class BV: the Gaussian case . . . . . . . . . . 8 The class BV: the general case . . . . . . . . . . . 9 Sobolev functions on domains and their extensions 10 BV functions on domains and their extensions . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

1

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. . . . . . . . . . .

. . . . . . . . . . .

Introduction to the Heisenberg group Hn . . . . . . . 1.1 Algebraic structure . . . . . . . . . . . . . . 1.2 Metric structure . . . . . . . . . . . . . . . . Heisenberg perimeter and other equivalent measures 2.1 H -perimeter . . . . . . . . . . . . . . . . . 2.2 Equivalent notions for H -perimeter . . . . . 2.3 RectiVability of the reduced boundary . . . . Area formulas, Vrst variation and H -minimal surfaces 3.1 Area formulas . . . . . . . . . . . . . . . . . 3.2 First variation and H -minimal surfaces . . . 3.3 First variation along a contact Wow . . . . . .

. . . . . . . . . . .

. . . . . . . . . . .

Roberto Monti Isoperimetric problem and minimal surfaces in the Heisenberg group

1

2

3

2 6 14 16 25 29 33 35 39 42 50

57

58 58 59 66 66 73 76 78 78 85 94

vi

4

Isoperimetric problem . . . . . . . . . . . . . . . . . . . 4.1 Existence of isoperimetric sets and Pansu’s conjecture . . . . . . . . . . . . . . . . . . . . . . . 4.2 Isoperimetric sets of class C 2 . . . . . . . . . . 4.3 Convex isoperimetric sets . . . . . . . . . . . . 4.4 Axially symmetric solutions . . . . . . . . . . . 4.5 Calibration argument . . . . . . . . . . . . . . . 5 Regularity problem for H -perimeter minimizing sets . . 5.1 Existence and density estimates . . . . . . . . . 5.2 Examples of nonsmooth H -minimal surfaces . . 5.3 Lipschitz approximation and height estimate . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

99 99 105 107 108 111 113 114 117 123 125

Emanuele Spadaro Regularity of higher codimension area minimizing integral currents 131

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . 132 1.1 Integer rectiVable currents . . . . . . . . . . . . 133 1.2 Partial regularity in higher codimension . . . . . 135 2 The blowup argument: a glimpse of the proof . . . . . . 136 3 Q-valued functions and rectiVable currents . . . . . . . 141 4 Selection of contradiction’s sequence . . . . . . . . . . . 147 5 Center manifold’s construction . . . . . . . . . . . . . . 151 5.1 Notation and assumptions . . . . . . . . . . . . 152 5.2 Whitney decomposition and interpolating functions153 5.3 Normal approximation . . . . . . . . . . . . . . 156 5.4 Construction criteria . . . . . . . . . . . . . . . 157 5.5 Splitting before tilting . . . . . . . . . . . . . . 159 5.6 Intervals of Wattening . . . . . . . . . . . . . . . 161 5.7 Families of subregions . . . . . . . . . . . . . . 162 6 Order of contact . . . . . . . . . . . . . . . . . . . . . . 164 6.1 Frequency function’s estimate . . . . . . . . . . 165 6.2 Boundness of the frequency . . . . . . . . . . . 175 7 Final blowup argument . . . . . . . . . . . . . . . . . . 176 7.1 Convergence to a Dir-minimizer . . . . . . . . . 178 7.2 Persistence of singularities . . . . . . . . . . . . 182 8 Open questions . . . . . . . . . . . . . . . . . . . . . . 188 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 Davide Vittone The regularity problem for sub-Riemannian geodesics

1 2

193

Introduction . . . . . . . . . . . . . . . . . . . . . . . . 193 The Carnot-Carath´eodory distance . . . . . . . . . . . . 195

vii

2.1 DeVnition of Carnot-Carath´eodory distance . . . 2.2 The Chow-Rashevski theorem . . . . . . . . . . 2.3 The Ball-Box Theorem . . . . . . . . . . . . . . 3 Length minimizers and extremals . . . . . . . . . . . . . 3.1 Length minimizers, existence and non-uniqueness 3.2 First-order necessary conditions . . . . . . . . . 3.3 Normal extremals . . . . . . . . . . . . . . . . . 3.4 Abnormal extremals . . . . . . . . . . . . . . . 3.5 An interesting family of extremals . . . . . . . . 4 Carnot groups . . . . . . . . . . . . . . . . . . . . . . . 4.1 StratiVed groups . . . . . . . . . . . . . . . . . 4.2 Carnot groups . . . . . . . . . . . . . . . . . . . 4.3 The dual curve and extremal polynomials . . . . 4.4 Extremals in Carnot groups . . . . . . . . . . . 5 Minimizers in step 3 Carnot group . . . . . . . . . . . . 6 On the negligibility of the abnormal set . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .

195 196 197 198 198 200 205 206 209 210 210 212 212 216 218 219 223

Preface

In 2013, a school on Geometric Measure Theory and Real Analysis, organized by G. Alberti, C. De Lellis and myself, took place at the Centro De Giorgi in Pisa, with lectures by V. Bogachev, R. Monti, E. Spadaro and D. Vittone. The lectures were so well-organized and up-to-date that we suggested publishing them as Lecture Notes. All lecturers kindly agreed to this project. The book presents in a friendly and unitary way many recent developments which have not previously appeared in book form. Topics include: in1nite-dimensional analysis, minimal surfaces and isoperimetric problems in the Heisenberg group, regularity of sub-Riemannian geodesics and the regularity theory of area-minimizing currents in any dimension and codimension.

Sobolev classes on infinite-dimensional spaces Vladimir I. Bogachev

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Measures on inVnite-dimensional spaces . . . . . . . . . . . . . . . . . . . . . . 2 2. Gaussian measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3. Integration by parts and differentiable measures . . . . . . . . . . . . . . . 14 4. Sobolev classes over Gaussian measures . . . . . . . . . . . . . . . . . . . . . . 16 5. Inequalities and embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 6. Sobolev classes over differentiable measures . . . . . . . . . . . . . . . . . 29 7. The class BV: the Gaussian case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 8. The class BV: the general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 9. Sobolev functions on domains and their extensions . . . . . . . . . . . . 39 10. BV functions on domains and their extensions . . . . . . . . . . . . . . . 42 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Introduction Sobolev classes of functions of generalized differentiability belong to the major analytic achievements in the XX century and have found impressive applications in the most diverse areas of mathematics. So it does not come as a surprise that their inVnite-dimensional analogs attract considerable attention. It was already at the end of the 60s and the beginning of the 70s of the last century that in the works of N. N. Frolov, Yu. L. DaletskiZX, L. Gross, M. Kr´ee, and P. Malliavin Sobolev classes with respect to Gaussian measures on inVnite-dimensional spaces were introduced and studied. Their Vrst triumph came with the development of the Malliavin calculus since the mid of the 70s. At present, such classes and their generalizations have become a standard tool of inVnite-dimensional anal-

This work was supported by the RSF project 14-11-00196.

2 Vladimir I. Bogachev

ysis. They Vnd applications in stochastic analysis, optimal transportation, mathematical physics, and mathematical Vnance. The aim of this survey is to give a concise account of the theory of Sobolev classes on inVnite-dimensional spaces with measures. We present a number of already classical cornerstone achievements, some more recent results, and open problems with relatively short formulations. There are already some books presenting elements of this rapidly developing theory (mostly in the Gaussian case), see Bogachev [13, 16] (see also [12]), Bouleau, Hirsch [25], Da Prato [34], Fang [41], Janson [58], Malliavin [66], Malliavin, Thalmaier [67], Nourdin, Peccati [73], Nu¨ unel, Zakai [87]. There is also another alart [74], Shigekawa [83], and Ust¨ direction developing Sobolev classes on the so-called measure metric spaces, see Ambrosio, Di Marino [7], Ambrosio, Tilli [10], Cheeger [30], Hajasz, Koskela [53], Heinonen [54], Keith [59], Reshetnyak [77–79], Vodop’janov [88], which is quite different from the topics discussed here. The survey is based on several courses I lectured at the Scuola Normale Superiore di Pisa in the years 1995–2013. Over the years I have had a splendid opportunity to discuss problems related to Sobolev classes in inVnite dimensions with many experts in this Veld, including H. Airault, L. Ambrosio, G. Da Prato, D. Elworthy, S. Fang, D. Feyel, M. Fukushima, M. Hino, A. Lunardi, P. Malliavin, P.-A. Meyer, D. Nualart, M. R¨ockner, I. Shigekawa, S. Watanabe, N. Yoshida, and M. Zakai.

1 Measures on inHnite-dimensional spaces Given a topological space X we denote by B(X) its Borel σ -Veld. Bounded measures on B(X) (possibly, signed) will be called Borel measures. Such a measure μ can be uniquely written as μ = μ+ − μ− , where μ+ and μ− are mutually singular nonnegative measures called the positive and negative parts of μ, respectively. Set |μ| = μ+ + μ− ,

μ = |μ|(X).

The class of all μ-integrable functions is denoted by L1 (μ) and the corresponding Banach space of equivalence classes (where functions equal almost everywhere are identiVed) is denoted by L 1 (μ). Similar notation L p (μ) and L p (μ) is used for the classes of μ-measurable functions integrable to power p ∈ (1, ∞) and the respective spaces of equivalence classes. For a Hilbert space H , the symbol L p (μ, H ) is used to denote the L p -space of H -valued mappings.

3 Sobolev classes on infinite-dimensional spaces

If a measure ν on B(X) has the form ν =  · μ, where  is a μintegrable function, which means that  ν(A) = (x) μ(dx), A ∈ B(X), A

then  is called absolutely continuous with respect to μ, which is denoted by ν  μ, and  is called its Radon–Nikodym density with respect to μ. A necessary and sufVcient condition for that, expressed by the Radon– Nikodym theorem, is that ν vanishes on all sets of μ-measure zero. If also μ  ν, which is equivalent to  = 0 μ-a.e., then the measures are called equivalent, which is denoted by ν ∼ μ. A nonnegative Borel measure μ on a topological space X is called Radon if, for every set B ∈ B(X) and every ε > 0, there is a compact set K ε ⊂ B such that μ(B\K ε ) < ε. Theorem 1.1. Each Borel measure on any complete separable metric space X is Radon. Moreover, this is true for any Souslin space X, i.e., the image of a complete separable metric space under a continuous mapping. In particular, this is true for the spaces C[0, 1], R∞ , and all separable Hilbert spaces. For the purposes of this survey it is sufVcient to have in mind the space ∞ R , the countable power of the real line with its standard product topology (making it a complete metrizable space). The Borel σ -Veld in this space coincides with the smallest σ -Veld containing all cylinders, i.e., sets of the form C B = {x : (x1 , . . . , xn ) ∈ B},

B ∈ B(Rn ).

The measure μn deVned on Rn by the formula μn (B) = μ(C B ) is called the projection of μ on Rn . These projections are consistent in the sense that the projection of μn+1 on Rn equals μn . By Kolmogorov’s theorem, the converse is true: given a consistent sequence of probability measures μn on the spaces Rn , there is a unique probability measure μ on R∞ with these projections (and there is a natural extension of this result to the case of signed measures, where in the inverse implication the uniform boundedness of μn is required). There is a dual concept to that of projections: conditional measures. Let us consider the one-dimensional subspace Re1 generated by the Vrst

4 Vladimir I. Bogachev

coordinate vector e1 and its natural complementing hyperplane Y1 consisting of vectors x with x1 = 0. Let ν 1 be the projection of |μ| to Y1 , i.e., its image under the natural projecting to Y1 . It is known (see [15, Chapter 10]) that there are Borel measures μ1,y , y ∈ Y1 , on the real line (these measures are probability measures if so is μ) such that for every bounded Borel function f , writing x as x = (x1 , y) with y = (x2 , x3 , . . .) and identifying y with (0, x2 , x3 , . . .), one has    f (x1 , x2 , . . .) μ(dx) = f (x1 , y) μ1,y (dx1 ) ν 1 (dy), where the function deVned by the integral in x1 is Borel measurable in y. Similarly, there exist conditional measures μn,y , y ∈ Yn , corresponding to the nth coordinate vector en and its natural complementing hyperplane Yn consisting of vectors with zero nth coordinate. Unlike Vnitedimensional distributions, conditional measures (even regarded for all n) do not uniquely determine the measure; the problem of reconstructing a measure from its conditional measures is the subject of the theory of Gibbs measures. Of course, it is not essential that we have considered basis vectors. For a general Radon measure μ (possibly, signed) on a locally vector space X that is a direct topological sum of two closed linear subspaces Z and Y , letting ν be the image of |μ| under the projection on Y , one can Vnd Radon measures μ y , y ∈ Y , on Z such that for each bounded Borel function f on X one has    f dμ = f (z, y) μ y (dz) ν(dy), X

Y

Z

where we write elements of X as x = (z, y), z ∈ Z , y ∈ Y , the function y → μ y  is ν-integrable, and the inner integral is also ν-integrable. Finally, note that sometimes it is more convenient geometrically to deVne the conditional measures μ y on the straight lines Rh + y rather than on the real line. In that case the previous equality reads simply as    f dμ = f (x) μ y (dx) ν(dy). X

Y

Z

It will be useful below to represent different measures μ and σ via conditional measures using a common measure ν on Y that may be different from their projections on Y . This is possible if take ν on Y such that both projections are absolutely continuous with respect to ν. Indeed, if μ = μ y μY (dy) and σ = σ y σY (dy), where μY = g1 · ν, σY = g2 · ν, then we obtain the representations μ = g1 (y)μ y ν(dy),

σ = g2 (y)σ y ν(dy)

5 Sobolev classes on infinite-dimensional spaces

or

μ = μ y,ν ν(dy),

σ = σ y,ν ν(dy),

where ν in the symbol μ y,ν indicates that the disintegration is taken with respect to the measure ν on Y in place of μY . We recall the deVnition of variation and semivariation of vector measures (see Diestel, Uhl [38] or Dunford, Schwartz [39]). Let H be a separable Hilbert space. A vector measure with values in H is an H valued countably additive function η deVned on a σ -algebra A of subsets of a space . Such a measure automatically has bounded semivariation deVned by the formula n     αi η(i ) , V (η) := sup i=1

H

where sup is taken over all Vnite partitions of  into disjoint parts i ∈ A and all Vnite sets of real numbers αi with |αi | ≤ 1. In other words, this is the supremum of variations of real measures (η, h) H over h ∈ H with |h| H ≤ 1. However, this does not yet mean that the vector measure η has Vnite variation which is deVned as Var(η) := sup

n 

|η(i )| H ,

i=1

where sup is taken over all Vnite partitions of  into disjoint parts i ∈ A. The variation of the measure η will be denoted by η (but in [39] this notation is used for semivariation). By the Pettis theorem (see Dunford, Schwartz [39, Chapter IV, §10]), an H -valued mapping is a vector measure of bounded semivariation provided that ( , h) H is a bounded scalar measure for each h ∈ H . The sets of measures of bounded variation and bounded semivariation are Banach spaces with the norms η → η and η → V (η), respectively. It is easy to give an example of a measure with values in an inVnite-dimensional Hilbert space having bounded semivariation, but inVnite variation: consider the standard basis {en } in l 2 and take Dirac’s measures δ(en ) in the points en and the vector measure η = ∞ −1 n δ(en )en . Its semivariation equals the sum of the numbers n −2 , n=1 but it is of inVnite variation. The space of all continuous linear functions on a locally convex space X is denoted by X ∗ and is called the dual (or topological dual) space. Let FC ∞ denote the class of all functions f on X of the form f (x) = f 0 (l1 (x), . . . , ln (x)),

f 0 ∈ Cb∞ (Rn ), li ∈ X ∗ ,

6 Vladimir I. Bogachev

where Cb∞ (Rn ) is the class of all inVnitely differentiable functions on Rn with bounded derivatives. In case of R∞ we obtain just the union of all Cb∞ (Rn ).

2 Gaussian measures A Gaussian measure on the real line is a Borel probability measure which is either concentrated at some  point a (i.e., is Dirac’s measure δa at a) or has density (2πσ )−1/2 exp −(2σ )−1 (x − a)2 with respect to Lebesgue measure, where a ∈ R1 is its mean and σ > 0 is its dispersion. The measure for which a = 0 and σ = 1 is called standard Gaussian. Similarly the standard Gaussian measure on Rd is deVned by its density (2π)−d/2 exp(−|x|2 /2) with respect to Lebesgue measure. Although below a general concept of a Gaussian measure on a locally convex space is introduced, we deVne explicitly general Gaussian measures on Rd . These are measures that are concentrated on afVne subspaces in Rd and are standard in suitable (afVne) coordinate systems. In other words, these are images of the standard Gaussian measure under afVne mappings of the form x → Ax + a, where A is a linear operator and a is a vector. A bit more explicit representation is provided by the Fourier transform of a bounded Borel measure μ on Rd deVned by the formula     μ(y) = exp i(y, x) μ(dx), y ∈ Rd . In these terms, a measure μ is Gaussian if and only if its Fourier transform has the form

1  μ(y) = exp i(y, a) − Q(y, y) , 2 where Q is nonnegative quadratic form on Rd . The Fourier transform of the standard Gaussian measure is given by γ (y) = exp(−|y|2 /2).  The change of variables formula yields the following relation between A and Q if μ is the image of γ under the afVne mapping Ax + a:     μ(y) = exp i(y, Ax + a) γ (dx)      = exp i(y, a) exp i(A∗ y, x) γ (dx)   = exp i(y, a) − |A∗ y|2 /2 ,

7 Sobolev classes on infinite-dimensional spaces

that is, Q(y) = (A A∗ y, y). It is readily veriVed that μ has a density on the whole space precisely when A is invertible. The vector a is called the mean of μ and is expressed by the equality  (y, a) = (y, x) μ(dx). For the quadratic form Q we have the equality  Q(y, y) = (y, x − a)2 μ(dx). These equalities are veriVed directly (it sufVces to check them in the onedimensional case). Let us deVne Gaussian measures on general locally convex spaces. DeHnition 2.1. Let X be a locally convex space with the topological dual X ∗ . A Borel probability measure γ on X is called Gaussian if the induced measure γ ◦ f −1 is Gaussian for every f ∈ X ∗ . If all these measures are centered, then γ is called centered. In the case of the space R∞ the space R∞ 0 of Vnite sequences coincides with the dual space. Hence Gaussian measures on R∞ are measures with Gaussian Vnite-dimensional projections. Example 2.2. An important example of a Gaussian measure is the countable product γ of the standard Gaussian measures on the real line. This measure is deVned on the space X = R∞ . This special example plays a very important role in the whole theory. In some sense (see Bogachev [13] for details) this is a unique up to isomorphism inVnite-dimensional Gaussian measure. Another important example of a Gaussian measure is the Wiener measure on the space C[0, 1] of continuous functions or on the space L 2 [0, 1]. This measure can be deVned as the image of the standard Gaussian measure γ on X = R∞ under the mapping  t ∞  xn en (s) ds, x = (xn ) → w( · ), w(t) = n=1

0

where {en } is an arbitrary orthonormal basis in L 2 [0, 1]. One can show that this series converges in L 2 [0, 1] for γ -almost every x; moreover, for γ -almost every x convergence is uniform on [0, 1].  We recall that a countable product μ = ∞ n=1 μn of probability measures μn on spaces (X n , Bn ) is deVned on X = ∞ n=1 X n as follows: Vrst it is deVned on sets of the form A = A1 × · · · × An × X n+1 · · · by μ(A) = μ1 (A1 ) × · · · × μn (An ),

8 Vladimir I. Bogachev

then it is veriVed that μ is countably additive on the algebra of Vnite unions of such sets (called cylindrical sets), which results ∞ in a countably additive extension to the smallest σ -algebra B := n=1 Bn containing such cylindrical sets. The standard Gaussian measure γ on R∞ can be restricted to many other smaller linear subspaces of full measure. For example, taking any sequence of numbers αn > 0 with ∞ n=1 αn < ∞, we can restrict γ to the weighted Hilbert space of sequences ∞

  E := (xn ) ∈ R∞ : αn xn2 < ∞ , n=1

making this expression the square of the norm. The fact that γ (E) = 1 follows by the monotone convergence theorem, which shows that ∞ 

αn xn2 < ∞

n=1

almost everywhere due to convergence of the integrals of the terms (the integral of xn2 is 1). Similarly, one can Vnd non-Hilbert full measure Banach spaces of sequences (xn ) with supn βn |xn | < ∞ or lim βn |xn | = 0 n→∞ for suitable sequences βn → 0; more precisely, the condition is this: C

exp − 2 < ∞ ∀ C > 0. βn n=1

∞ 

However, there is no minimal linear subspace of full measure. The point is that the intersection of all linear subspaces of positive (equivalently, full) measure is the subspace l 2 , which has measure zero, as one can verify directly. It is known that any Radon Gaussian measure γ has mean m ∈ X, i.e., m is a vector in X such that  f (x) γ (dx) ∀ f ∈ X ∗ . f (m) = X

If m = 0, i.e., the measures γ ◦ f −1 for f ∈ X ∗ have zero mean, then γ is called centered. Any Radon Gaussian measure γ is a shift of a centered Gaussian measure γm deVned by the formula γm (B) := γ (B + m). Hence for many purposes it sufVces to consider only centered Gaussian measures. For a centered Radon Gaussian measure γ we denote by X γ∗ the closure of X ∗ in L 2 (γ ). The elements of X γ∗ are called γ -measurable linear

9 Sobolev classes on infinite-dimensional spaces

functionals. There is an operator Rγ : X γ∗ → X, called the covariance operator of the measure γ , such that  f (x)g(x) γ (dx) ∀ f ∈ X ∗ , g ∈ X γ∗ . f (Rγ g) = X

Set

g :=  h if h = Rγ g.

Then  h is called the γ -measurable linear functional generated by h. The following vector equality holds (if X is a Banach space, then it holds in Bochner’s sense):  Rγ g = g(x)x γ (dx) ∀ g ∈ X γ∗ . X

For example, if γ is a centered Gaussian measure on a separable Hilbert space X, then there exists a nonnegative nuclear operator K on X for which K y = Rγ y for all y ∈ X, where we identify X ∗ with X. Then we obtain   γ (y) = exp −(K y, y)/2 . (K y, z) = (y, z) L 2 (γ ) and  Let us take an orthonormal eigenbasis {en } of the operator K with eigenvalues {kn }. Then γ coincides with the image of the countable power γ0 of the standard Gaussian measure on R1 under the mapping R∞ → X,

(xn ) →

∞   kn xn en . n=1

 2 This series converges γ0 -a.e. in X by convergence of the series ∞ n=1 kn x n , which follows by convergence of the series of kn and the fact that the integral of xn2 against the measure γ0 equals 1. Here X γ∗ can be identiVed √ with the completion of X with respect to the norm x →  K x X , i.e., the embedding X = X ∗ → X γ∗ is a Hilbert–Schmidt operator. The space H (γ ) = Rγ (X γ∗ ) is called the Cameron–Martin space of the measure γ . It is a Hilbert space with respect to the inner product  (h, k) H :=  h(x) k(x) γ (dx). X

The corresponding norm is given by the formula |h| H :=  h L 2 (γ ) .

10 Vladimir I. Bogachev

Moreover, it is known that H (γ ) with the indicated norm is separable and its closed unit ball is compact in the space X. Note that the same norm is given by the formula   |h| H = sup f (h) : f ∈ X ∗ ,  f  L 2 (γ ) ≤ 1 .   It should be noted that if dim H (γ ) = ∞, then γ H (γ ) = 0. In terms of the inner product in H the vector Rγ (l) is determined by the identity    f g dγ , f ∈ X ∗ , g ∈ X γ∗ . (2.1) jH ( f ), Rγ g H = f (Rγ g) = X

In the above example of a Gaussian measure γ on a Hilbert space we have √ H (γ ) = K (X). Let us observe that H (γ ) coincides also with the set of all vectors of the form  f (x)x γ (dx), f ∈ L 2 (γ ). h= X

Indeed, letting f 0 be the orthogonal projection of f onto X γ∗ in L 2 (γ ), we see that the integral of the difference [ f (x) − f 0 (x)]x over X vanishes since the integral of [ f (x) − f 0 (x)]l(x) vanishes for each l ∈ X ∗ . Theorem 2.3. The mapping h →  h establishes a linear isomorphism between H (γ ) and X γ∗ preserving the inner product. In addition, Rγ  h = h. en } is an orthonormal If {en } is an orthonormal basis in H (γ ), then { basis in X γ∗ and en are independent random variables. One can take an orthonormal basis in X γ∗ consisting of elements ξn ∈ X ∗ . The general form of an element l ∈ X γ∗ is this: l=

∞ 

cn ξn ,

n=1

where the series converges in L 2 (γ ). Since ξn are independent Gaussian random variables, this series converges also γ -a.e. The domain of its convergence is a Borel linear subspace L of full measure. One can take a version of l which is linear on all of X in the usual sense; it is called a proper linear version. It is easy to show that such a version is automatically continuous on H (γ ) with the norm | · | H ; more precisely,  f 0 (h) = (Rγ f, h) H = f h dγ , h ∈ H. X

11 Sobolev classes on infinite-dimensional spaces

Conversely, any continuous linear functional l on the Hilbert space H (γ ) admits a unique extension to a γ -measurable proper linear functional  l such that  l coincides with l on H (γ ). For every h ∈ H (γ ),  such an extension of the ∞functional x → (x, h) H is exactly h. If h = ∞   n=1 cn en , then h = n=1 cn en . Two γ -measurable linear functionals are equal almost everywhere precisely when their proper linear versions coincide on H (γ ). If a measure γ on X = R∞ is the countable power of the standard Gaussian measure on the real line, then X ∗ can be identiVed with the space of all sequences of the form f = ( f 1 , . . . , f n , 0, 0, . . .). Here we have ∞  f i gi . ( f, g) L 2 (γ ) = i=1

X γ∗

Hence can be identiVed with l ; any element l = (cn ) ∈ l 2 deVnes  an element of L 2 (γ ) by the formula l(x) := ∞ n=1 cn x n , where the series converges in L 2 (γ ). Therefore, the Cameron–Martin space H (γ ) coincides with the space l 2 with its natural inner product. An element l represents a continuous linear functional precisely when only Vnitely many numbers cn are nonzero. For the Wiener measure on C[0, 1] the Cameron–Martin space coincides with the class W02,1 [0, 1] of all absolutely continuous functions h on [0, 1] such that h(0) = 0 and h  ∈ L 2 [0, 1]; the inner product is given by the formula 2

 (h 1 , h 2 ) H :=

1

0

h 1 (t)h 2 (t) dt.

The next classical result, called the Cameron–Martin formula, relates measurable linear functionals and vectors in the Cameron–Martin space to the Radon–Nikodym density for shifts of the Gaussian measure. Theorem 2.4. The space H (γ) is the set of all h ∈ X such that γh ∼ γ , where γh (B) := γ (B + h), and the Radon–Nikodym density of the measure γh with respect to γ is given by the following Cameron–Martin formula:   dγh /dγ = exp − h − |h|2H /2 . For every h  ∈ H (γ ) we have γ ⊥ γh . It follows from this formula that for every bounded Borel function f on X we have     f (x + h) γ (dx) = f (x) exp  h(x) − |h|2H /2 γ (dx). X

X

12 Vladimir I. Bogachev

In the case of the standard Gaussian measure on R∞ this formula is a straightforward extension of the obvious Vnite-dimensional expression, one just needs to deVne  h(x) as the sum of a series. A centered Radon Gaussian measure is uniquely determined by its Cameron–Martin space (with the indicated norm!): if μ and ν are centered Radon Gaussian measures such that H (μ) = H (ν) and |h| H (μ) = |h| H (ν) for all h ∈ H (μ) = H (ν), then μ = ν. The Cameron-Martin space is also called the reproducing Hilbert space. DeHnition 2.5. A Radon Gaussian measure γ on a locally convex space X is called nondegenerate if for every nonzero functional f ∈ X ∗ the measure γ ◦ f −1 is not concentrated at a point. The nondegeneracy of γ is equivalent to that γ (U ) > 0 for all nonempty open sets U ⊂ X. This is also equivalent to that the CameronMartin space H (γ ) is dense in X. For every degenerate Radon Gaussian measure γ there exists the smallest closed linear subspace L ⊂ X for which γ (L + m) = 1, where m is the mean of the measure γ . Moreover, L + m coincides with the topological support of γ . If m = 0, then on L the measure γ is nondegenerate. Let γ be a centered Radon Gaussian measure on a locally convex space X; as usual, one can assume that this is the standard Gaussian measure on R∞ . The Ornstein–Uhlenbeck semigroup is deVned by the formula     f e−t x − 1 − e−2t y γ (dy), f ∈ L p (γ ). (2.2) Tt f (x) = X

A simple veriVcation of the fact that {Tt }t≥0 is a strongly continuous semigroup on all L p (γ ), 1 ≤ p < ∞, can be found in [13]; the semigroup property means that Tt+s f = Ts Ts f,

t, s ≥ 0.

An important feature of this semigroup is that the measure γ is invariant for it, that is,   Tt f (x) γ (dx) = X

f (x) γ (dx). X

Theorem 2.6. For every p ∈ [1, +∞) and f ∈ L p (γ ) one has       lim Tt f − f  L p (γ ) = 0, lim Tt f − f dγ  =0  t→+∞

t→0

and if 1 < p < ∞, then also lim Tt f (x) = f (x) a.e. t→0

L p (γ )

13 Sobolev classes on infinite-dimensional spaces

It is also known that in the Vnite-dimensional case lim Tt f (x) = f (x) t→0

a.e. for all f ∈ L 1 (γ ). It remains an open problem whether this is true in inVnite dimensions. The generator L of the Ornstein–Uhlenbeck semigroup is called the Ornstein–Uhlenbeck operator (more precisely, for every p ∈ [1, +∞), there is such a generator on the corresponding domain in L p (γ ); if p is not explicitly indicated, then usually p = 2 is meant). By deVnition, L f = lim(Tt f − f )/t if this limit exists in the norm of L p (γ ). This t→0

operator will be important Section 4. In the case of R∞ , on smooth functions f (x) = f (x1 , . . . , xn ) in Vnitely many variables one can explicitly calculate that n  2     ∂xi f (x) − xi ∂xi f (x) . L f (x) =  f (x) − x, ∇ f (x) = i=1

This representation can be also extended to some functions in inVnitely many variables. In the general case L f is the sum of a similar series, but its two parts need converge separately. In the theory of Gaussian measures an important role is played by the Hermite (or Chebyshev–Hermite) polynomials Hn deVned by the equalities (−1)n 2 d n  2  H0 = 1, Hn (t) = √ et /2 n e−t /2 , n > 1. dt n! They have the following properties: √ √ Hn (t) = n Hn−1 (t) = t Hn (t) − n + 1Hn+1 (t). In addition, the system of functions {Hn } is an orthonormal basis in L 2 (γ ), where γ is the standard Gaussian measure on the real line. For the standard Gaussian measure γn on Rn (the product of n copies of the standard Gaussian measure on R1 ) an orthonormal basis in L 2 (γn ) is formed by the polynomials of the form Hk1 ,...,kn (x1 , . . . , xn ) = Hk1 (x1 ) · · · Hkn (xn ),

ki ≥ 0.

If γ is a centered Radon Gaussian measure on a locally convex space X and {ln } is an orthonormal basis in X γ∗ , then a basis in L 2 (γ ) is formed by the polynomials     Hk1 ,...,kn (x) = Hk1 l1 (x) · · · Hkn ln (x) , ki ≥ 0, n ∈ N. For example, for the countable power of the standard Gaussian measure on the real line such polynomials are Hk1 ,...,kn (x1 , . . . , xn ). It is

14 Vladimir I. Bogachev

convenient to arrange polynomials Hk1 ,...,kn according to their degrees k1 + · · · + kn . For k = 0, 1, . . . we denote by Xk the closed linear subspace of L 2 (γ ) generated by the functions Hk1 ,...,kn with k1 +· · ·+kn = k. The functions Hk1 ,...,kn are mutually orthogonal and, for the Vxed value k = k1 + · · · + kn , form an orthonormal basis in Xk . The one-dimensional space X0 consists of constants and X1 = X γ∗ . One can show that every element f ∈ X2 can be written in the form f =

∞ 

αn (ln2 − 1),

n=1

 2 where {ln } is an orthonormal basis in X γ∗ and ∞ n=1 αn < ∞ (i.e., the series for f converges in L 2 (γ )). The spaces Xk are mutually orthogonal and their orthogonal sum is the whole L 2 (γ ): ∞  Xk , L 2 (γ ) = k=0

which means that, denoting by Ik the operator of orthogonal projection onto Xk , we have an orthogonal decomposition F=

∞ 

Ik (F),

F ∈ L 2 (γ ).

k=0

One can check that Tt Hk1 ,...,kn = e−k1 −···−kn Hk1 ,...,kn , which yields that Tt F =

∞ 

e−kt Ik (F),

F ∈ L 2 (γ ).

k=0

Given a separable Hilbert space E, one deVnes similarly the space Xk (E) of polynomials with values in E as the closure in L 2 (γ , E) of the liner span of the mappings f · v, where f ∈ Xk , v ∈ E.

3 Integration by parts and differentiable measures Suppose that f is a bounded Borel function on a locally convex space X with a centered Radon Gaussian measure γ such that the partial derivative f (x + th) − f (x) t→0 t exists for some vector h in the Cameron-Martin space of γ and is bounded. Applying the Cameron-Martin formula and Lebesgue’s dominated convergence theorem, we arrive at the equality   ∂h f (x) γ (dx) = f (x) h(x) γ (dx), ∂h f (x) = lim

X

X

15 Sobolev classes on infinite-dimensional spaces

 2 2 where we also use that the derivative of t → et h−t |h| H /2 at zero is  h. This simple formula, called the integration by parts formula for the Gaussian measure, plays a very important role in stochastic analysis and is a starting point for far-reaching generalizations connected with differentiabilities of measures in the sense of Fomin [45, 46] and in the sense of Skorohod [85]. A measure μ on X is called Skorohod differentiable along a vector h if there exists a measure dh μ, called the Skorohod derivative of the measure μ along the vector h, such that   f (x − th) − f (x) f (x)dh μ(dx) (1) lim μ(dx) = t→0 X t X

for every bounded continuous function f on X. If the measure dh μ is absolutely continuous with respect to the measure μ, then the measure μ is called Fomin differentiable along the vector h, the Radon–Nikodym density of the measure dh μ with respect to μ is denoted by βhμ and called the logarithmic derivative of μ along h. The Skorohod differentiability of μ along h is equivalent to the identity   ∂h f (x) μ(dx) = − f (x) dh μ(dx), f ∈ FC ∞ . X

X

The Fomin differentiability is the equality   ∂h f (x) μ(dx) = − f (x) βhμ (x) μ(dx), X

f ∈ FC ∞ .

X

On the real line the Fomin differentiability is equivalent to the membership of the density in the Sobolev class W 1,1 , and the Skorohod differentiability is the boundedness of variation of the density; the picture is similar also in Rn . A detailed discussion of these types of differentiability of measures can be found in Bogachev [16]. It follows from our previous discussion that for the centered Gaussian measure μ we have h, βhμ = −

h ∈ H (μ).

In the case of a probability measure on R∞ efVcient conditions for both types of differentiability can be expressed in terms of Vnite-dimensional distributions. The Skorohod differentiability along a vector h = (h n ) is equivalent to the following condition: for every n, the generalized derivative of the projection μn on Rn along the vector (h 1 , . . . , h n ) is a bounded measure and such measures are uniformly bounded. For Fomin’s differentiability more is needed: the corresponding logarithmic derivatives

16 Vladimir I. Bogachev

β μn (h 1 , . . . , h n ) are uniformly integrable with respect to μ (regarded as functions on R∞ ). An equivalent characterization is available in terms of conditional measures (see Section 1): Fomin’s differentiability of μ along h is equivalent to the following: the conditional measures μ y on the real line have densities  y ∈ W 1,1 (R) such that the function y → ∂t  y  L 1 (R) is μY integrable, where μY is the projection of μ on a closed hyperplane Y complementing the one-dimensional subspace generated by h. The derivative dh μ can be written as   ∂t  y (t) dt μY (dy), dh μ(B) = Y

By

where B y = {t ∈ R : (th, y) ∈ B}, B ∈ B(X), and X is written as Rh × Y . It is worth mentioning that for a Gaussian measure, the conditional measures are Gaussian as well (see Bogachev [13, Section 3.10] or [17]). It is known (see Bogachev [16, Chapter 5]) that the sets DC (μ) and D(μ) of all vectors of differentiability of a nonzero measure μ in the sense of Skorohod and Fomin respectively are Banach spaces with respect to the norm h → dh μ and that the closed unit ball in DC (μ) is compact in X. For any h ∈ D(μ) we have dh μ = βh  L 1 (μ) . For example, if μ is a Gaussian measure (as before, we need only the countable power of the Gaussian measure on the real line), then the set D(μ) = DC (μ) coincides with the Cameron–Martin space H (μ) of the measure μ (the set of all vectors the shifts to which give equivalent measures). We assume further that the measure μ is Fomin differentiable along all vectors in a separable Hilbert space H that is continuously and densely embedded into X (the model example is l 2 ⊂ R∞ ). Hence the closed graph theorem yields that the natural embedding H → D(μ) is continuous and dh μ ≤ C|h| H , h ∈ H for some constant C.

4 Sobolev classes over Gaussian measures In this section we brieWy discuss Sobolev spaces with respect to Gaussian measures. This is a very important analytical tool and one of the mainstreams in modern theory. The reason why such classes are important is that many nonlinear functionals on inVnite-dimensional spaces arising in applications have very poor differentiability or even continuity properties from the point of view of the classical analysis (norm continuity,

17 Sobolev classes on infinite-dimensional spaces

Fr´echet or Gˆateaux differentiability), but are Sobolev smooth. This effect is much stronger than in the Vnite-dimensional case (where it is also notable, e.g., in the theory of partial differential equations), and it was Paul Malliavin [65] who invented special tools (now called the Malliavin calculus) to deal with such problems. It should be noted that important ideas closely connected with Gaussian Sobolev classes were developed already by Gross [51] and the Vrst deVnition of such classes was given by Frolov [47, 48]. Later such classes were studied in DaletskiZX, Paramonova [35–37], Kr´ee [61, 62], Lascar [63], and in many other works. Similarly to the classical Sobolev spaces (see, e.g., the books Adams, Fournier [1], Lieb, Loss [64], Ziemer [92]), there are essentially three different ways of introducing such spaces: as suitable completions of smooth functions, in terms of integration by parts, and through integral representations. For example, the class W 1,1 (Rd ) can be deVned either as the completion of the class C0∞ of smooth compactly supported functions with respect to the Sobolev norm  f 1,1 =  f 1 + ∇ f 1 , where  · 1 denotes the L 1 -norm of scalar or vector functions, or the subclass in L 1 (Rd ) consisting of the functions whose generalized Vrst order partial derivatives belong to L 1 (Rd ), where the generalized partial derivative ∂xi f is deVned by means of the integration by parts formula  Rd

 ϕ∂xi f dx = −

Rd

f ∂xi ϕ dx,

ϕ ∈ C0∞ .

Using the L p -norm we arrive at the classes W p,1 (Rd ). Similar constructions work in the case of weighted Sobolev classes W p,1 (), where  is a nonnegative locally integrable function, so that in place of Lebesgue measure we use the measure μ =  dx. However, in this case some subtleties appear (see, e.g. Bogachev [16]). First of all, some conditions on  are needed to ensure the closability of the Sobolev norm, i.e., the property that if a sequence of smooth functions f j converges to zero in L p and is fundamental in the Sobolev norm, then it also converges to zero in the Sobolev norm. Next, the use of the integration by parts formula also imposes restrictions on  in the second approach. Finally, these two and other approaches may lead to distinct Sobolev classes unlike the classical case, see Zhikov [90], [91]. We Vrst consider the case of the standard Gaussian measure γ on Rd . The classes W p,1 (γ ), 1 ≤ p < ∞, are obtained as the completions of the

18 Vladimir I. Bogachev

class C0∞ (Rd ) with respect to the Sobolev norms   f  p,1 :=

1/ p | f | p dγ

 +

1/ p |∇ f | p dγ

.

Similarly one deVnes the classes W p,1 (γ , Rm ) of Rm -valued Sobolev mappings. An extension to higher order derivatives is relatively straightforward, but there is a nuance in the choice of the norm on higher order derivatives: for many purposes it turns out to be reasonable to take Hilbert–Schmidt norms (rather than other matrix norms). In particular, the space W p,2 (γ ) is obtained by taking the norm  f  p,2 :=  f  p,1 +

 

|∂xi ∂x j f |2

p/2

1/ p dγ

.

i, j≤d

Continuing inductively we obtain the spaces W p,r (γ ), r ∈ N. The same class W p,r (γ ) is characterized as follows: it consists of all functions f ∈ L p (γ ) such that f possesses generalized partial derivatives ∂xi1 · · · ∂xir f represented by elements in L p (γ ). The inVnite-dimensional case, where γ is a centered Radon Gaussian measure on a locally convex space with the Cameron–Martin space H , is completely analogous, the only difference is that now in place of C0∞ we take the class FC ∞ of all functions on X of the form   f (x) = f 0 l1 (x), . . . , ln (x) ,

li ∈ X ∗ , f 0 ∈ Cb∞ (Rn ).

Let {ei } be an orthonormal basis in H . Set ∂h f (x) = lim

t→∞

f (x + th) − f (x) . t

For all p ≥ 1 and r ∈ N, the Sobolev norm  · W p,r is deVned by the following formula, where ∂i := ∂ei :  f W p,r

1/ p  r      2 p/2 ∂i1 . . . ∂ik f (x) = γ (dx) . k=0

(4.1)

X i 1 ,...,i k ≥1

If X = R∞ and H = l 2 , then FC ∞ is just the space of functions of Vnitely many variables of class Cb∞ and if γ is the standard Gaussian measure on R∞ , then the Sobolev norms on such functions are the previously deVned norms in the Vnite-dimensional case.

19 Sobolev classes on infinite-dimensional spaces

Let W p,r (γ ) denote the completion of FC ∞ with respect to the Sobolev norm  ·  p,r =  · W p,r . Note that the same norm can be written as r  D kH f  L p (γ ,Hk ) ,  f  p,r = k=0

where f stands for the derivative of order k along H and Hk is the space of Hilbert–Schmidt k-linear forms on H , which can be deVned inductively by setting Hk = H(H, Hk−1 ), H1 = H , where H(H, E) is the space of Hilbert–Schmidt operators between Hilbert spaces H and E equipped with its natural norm deVned by D kH

T 2H =

∞ 

T ei 2E

i=1

for an arbitrary orthonormal basis {ei } in H . After this completion procedure all elements in W p,r (γ ) acquire Sobolev derivatives D kH f of the respective orders. In particular, any f ∈ W 2,1 (γ ) has a Sobolev gradient D H f along H , which is a limit in L 2 (γ , H ) of the H -gradients of smooth cylindrical functions convergent to f in the norm , · 2,1 . For example, in the case of the standard measure on R∞ the ∞ Gaussian −1 measurable linear functional mf (x)−1= n=1 n xn belongs to all classes p,r W (γ ), since the sums n=1 n xn converge in each norm  ·  p,r ; )∞ more speciVcally, D H f (x) is a constant vector h = (n −1 n=1 and all −2 2 higher derivatives vanish. Similarly, the function f (x) = ∞ n=1 n x n ∞ 2 p,r −2 belongs to all classes W (γ ), D H f (x) = 2(n xn )n=1 , D H f (x) is constant and equals the diagonal operator with eigenvalues 2n −2 , higher order derivatives vanish. In a similar way one deVnes the Sobolev spaces W p,r (γ , E) of mappings with values in a Hilbert space E. The corresponding norms are denoted by the same symbol || · || p,r . An equivalent description employs the concept of a Sobolev derivative. Let p > 1. We shall say that a function f ∈ L p (γ ) has a generalized (or Sobolev) partial derivative g ∈ L 1 (γ ) along a vector h ∈ H if, for every ϕ ∈ FC ∞ , one has the equality   ∂h ϕ(x) f (x) γ (dx) = − ϕ(x) g(x) γ (dx) X X (4.2)  + ϕ(x) f (x) h(x) γ (dx). X

Set ∂h f := g. Similarly one deVnes generalized partial derivatives for mappings with values in a separable Hilbert space E.

20 Vladimir I. Bogachev

DeHnition 4.1. Let p ∈ (1, +∞). The class G p,1 (γ ,E) consists of all  mappings f ∈ L p (γ ,E) such that there is a mapping D f ∈ L p γ , H(H,E) with the property that, for every h ∈ H , the E-valued mapping x → D f (x)h serves as a generalized partial derivative of f along h. The classes G p,r (γ , E) with r ∈ N are deVned inductively as follows: p,1 the class D p,r+1 (γ , E) consists of  all mappings  f ∈ G (γ , E) such p,r γ , Hr (H, E) , deVned at the previous that D f belongs to the class G inductive step, and the derivative of order r + 1 is deVned by Dr+1 H f = DrH D H f . Theorem 4.2. One has G p,r (γ , E) = W p,r (γ , E) if p ∈ (1, +∞), r ∈ N. Remark 4.3. The case p = 1 requires a special examination, since in the deVnition of generalized derivatives we used the fact that  h f ∈ L 1 (γ ), p which is true by H¨older’s inequality for any f ∈ L (γ ) with p > 1. The space G 1,1 (γ ) can be deVned as the space of all functions f ∈ L 1 (γ ) such that  h f ∈ L 1 (γ ) for all h ∈ H and there is a mapping D H f ∈ L 1 (γ , H ) for which the function (D H f, h) H serves as a generalized partial derivative along h for each h ∈ H . However, as shall see in the lemma below, the inclusion  h f ∈ L 1 (γ ) is automatically fulVlled if f has a directional partial derivative ∂h f ∈ L 1 (γ ) in the sense considered below. A closely related description focuses on directional properties of functions in the Sobolev classes. We present a typical result for r = 1 and E = R1 ; extensions to greater r and inVnite-dimensional E are straightforward. Let us Vx an orthonormal basis {ei } in H . Theorem 4.4. A function f in L p (γ ), p ≥ 1, belongs to W p,1 (γ ) pref such that the functions t → cisely when, for each ei , it has a version   f (x + tei ), where x ∈ X, are locally absolutely continuous and, setting ∂ei f (x) :=

d  f (x + tei )|t=0 , dt

∞ belonging to L p (γ , H ). The same we obtain a mapping ∇ f = (∂ei f )i=1 p,1 is true for the class G (γ ), so that W p,1 (γ ) = G p,1 (γ ) also for p = 1.

It should be added that the partial derivative ∂ei f (x) exists almost everywhere, since t →  f (x + tei ) is almost everywhere differentiable on the real line (by a classical result from real analysis), which yields through conditional measures that the derivative at zero exists for almost every Vxed x; certainly, for a given x there might be no derivative at zero. The reader is warned that a version  f with the required properties depends in general on ei , which is suppressed in our notation. This happens already in dimension 2: taking a function f ∈ W 2,1 (γ ) such that

21 Sobolev classes on infinite-dimensional spaces

every version of it is locally unbounded (it is easy to give an example), we see that f has no version continuous in each variable separately (such a version would have a point of continuity). Lemma 4.5. There is a constant C with the following property: if a function f in L 1 (γ ) has a version that is locally absolutely continuous on the lines x + R1 h for some h ∈ H and ∂h f ∈ L 1 (γ ), where ∂h f is deFned almost everywhere through the indicated version, then    (4.3) | h|| f | dγ ≤ C  f  L 1 (γ ) + ∂h f  L 1 (γ ) . Proof. In the one-dimensional case the assertion is obvious, because the integral of t| f (t)| over [0, ∞) with respect to the  standard Gaussian measure is estimated by C  f  L 1 (γ ) +  f   L 1 (γ ) with some constant C as follows. Let us deal with a locally absolutely continuous version of f . Then g  (t) = −tg(t) and by the integration by parts formula we have 



R

t| f (t)|g(t) dt =

0

0

R

R | f (t)| g(t) dt − | f (t)|g(t)0 ,

  whence, taking into account that | f (t)|  = | f (t) | a.e., we Vnd that 

+∞

−∞

|t|| f (t)|g(t) dt ≤ 2 f   L 1 (γ ) + | f (0)|.

Let us estimate f (0). We may assume that f (0) > 0. Let us take T > 0 such that [0, T ] has γ -measure 1/4. Next, we choose τ ∈ [0, T ] such that f (τ ) ≤ 4 f  L 1 (γ ) . Then, letting C1−1 := mint∈[0,T ] g(t), we have f (0) ≤ f (τ ) +  f   L 1 [0,τ ] ≤ 4 f  L 1 (γ ) + C1  f   L 1 (γ ) , so that 

+∞

−∞

  |t|| f (t)|g(t) dt ≤ C  f  L 1 (γ ) +  f   L 1 (γ ) ,

(4.4)

where C = 6 + C1 does not depend on f . The general case follows from this special one. Indeed, we can assume that |h| H = 1. Then the conditional measures γ x on the straight lines x + R1 h are standard Gaussian, which yields estimate (4.3). In fact, this can be seen even without conditional measures. The claim reduces to the case where γ is the standard product measure and h = e1 . Then it sufVces to use Fubini’s theorem and (4.4) for the Vrst coordinate and Vxed other coordinates.

22 Vladimir I. Bogachev

Yet another description of Sobolev classes (even with fractional orders of differentiability) employs the Ornstein-Uhlenbeck semigroup {Tt }. Let r > 0. Set  ∞ −1 t r/2−1 e−t Tt f dt, f ∈ L p (γ ), Vr f := (r/2) 0

where



∞

(α) :=

t α−1 e−t dt.

0

By the same formula we deVne Vr on L p (γ , E), where E is any separable Hilbert space. For p ≥ 1 and r > 0 let us consider the space   H p,r (γ ) := Vr L p (γ ) ,  f  H p,r = Vr−1 f  L p (γ ) . It is not difVcult to show that this space is complete. Let us note a common useful property of the classes of any of the three types with p > 1: if functions f n belonging to one of them converge in measure to a function f and supn  f n  p,r < ∞, then f belongs to the same class. Another common feature is the reWexivity of these spaces (which follows by the reWexivity of L p with 1 < p < ∞). It is very important that the derivatives in these constructions are taken along the subspace H , so that the geometry of the space X carrying the measure γ is irrelevant. If X itself is a nice space (say, Hilbert or Banach), then smooth functions in the classical Fr´echet or Gˆateaux sense with appropriate bounds on derivatives become Sobolev differentiable. However, no values of p and r ensure continuity of elements in W p,r . Example 4.6. Let γ be the standard Gaussian measure on R∞ restricted to the full measure Hilbert space E of sequence (xn ) with ∞ −2 n xn2 < ∞. Let n=1 ∞ 

f (x) =

n −2/3 xn .

n=1

Then the function f has no version continuous on E with its Hilbert norm, but f ∈ W p,r (γ ) for all r ∈ N and p ∈ [1, +∞), moreover, D H f is a constant vector and D kH f = 0 if k ≥ 2. A similar effect is seen in the case of the stochastic integral  f (w) = 0

1

ψ(t) dw(t),

23 Sobolev classes on infinite-dimensional spaces

where ψ ∈ L 2 [0, 1] has unbounded variation (say, just has no bounded version). Such stochastic integrals regarded as measurable linear functionals on C[0, 1] are given by continuous functionals on C[0, 1] (are represented as integrals of paths with respect to bounded measures) precisely when ψ is a function of bounded variation as an equivalence class in L 2 [0, 1], that is, has a modiVcation of bounded variation, see Bogachev [13, Problem 2.12.32]. For integer values of r the spaces H p,r (γ ) can be compared with the previously deVned classes. The following very important result is Meyer’s equivalence. Theorem 4.7. If p ∈ (1, +∞), r ∈ N, then H p,r (γ ) = W p,r (γ ) = D p,r (γ ) and there exist positive constants m p,r and M p,r such that m p,r DrH f  L p (γ ,Hr ) ≤ (I − L)r/2 f  L p (γ ) ≤   ≤ M p,r DrH f  L p (γ ,Hr ) +  f  L p (γ ) .

(4.5)

The same is true for E-valued mappings, where E is a separable Hilbert space. Let us observe that for any function f ∈ W p,2 (γ ) we have its second derivative D 2H f and the action L f of the Ornstein–Uhlenbeck operator on it. In the case of the standard Gaussian measure on Rn one has n  2     ∂xi f (x) − xi ∂xi f (x) , L f (x) =  f (x) − x, ∇ f (x) = i=1

  where both parts  f (x) = trace D 2 f (x) and x, ∇ f (x) exist separately. The same holds in the case of R∞ for functions of Vnitely many variables. However, for general functions f ∈ W 2,2 (γ ) in inVnite dimensions this is not example, let us consider a function f ∈ X2 given by true. For −1 2 n (x f (x) = ∞ n − 1). Then n=1 D H f (x) = 2(n −1 xn )∞ n=1 and

D 2H f (x) = A

is a constant Hilbert–Schmidt operator deVned by the diagonal matrix with the numbers 2n −1 at the diagonal. We have L f (x) = 2

∞  n=1

n −1 (1 − xn2 ),

24 Vladimir I. Bogachev

where the series converges in L 2 (γ ) and almost everywhere, but the part “ f ”, the trace of the second derivative, which is the series of n −1 , does not exist separately. Let us also note the following estimate (see, e.g., Shigekawa [83, Proposition 4.5] for the proof). Proposition 4.8. Let p > 1 and k ∈ N. Then there is a number C( p, k) such that D kH f  L p (γ ,Hk ) ≤ C( p, k)D k+1 H f  L p (γ ,Hk+1 ) + C( p, k) f  p for all f ∈ W p,k+1 (γ ). An analogous estimate holds for mappings f ∈ W p,k+1 (γ , E) with values in a separable Hilbert space E. A multiplicative inequality for Sobolev norms is ensured by the following result (see, e.g., Shigekawa [83, Proposition 4.10] for the proof). Proposition 4.9. Let α < β < κ. Then there is a number C(α, β, κ) > 0 such that κ−β

β−α

(I − L)β f  p ≤ C(α, β, κ)(I − L)α f  pκ−α (I − L)κ f  pκ−α for all f ∈ H p,κ (γ ).

For example, as a special case of this estimate (in fact, obtained as a step of the proof) one can obtain that 2 1/2 (I − L) f  p ≤ 2 f 1/2 p (I − L) f  p ,

which can be written as 1/2

 f  p,2 ≤ 2 f 1/2 p  f  p,4 . With a suitable number c we also have 1/2

 f  p,1 ≤ c f 1/2 p  f  p,2 . Sobolev functions satisfy certain vector integration by parts formulas. Theorem 4.10. Suppose that v ∈ W p,1 (γ , H ), where p > 1. Then there is a function δv ∈ L p (γ ), called the divergence of v, such that   (D H ϕ, v) H dγ = δvϕ dγ , ϕ ∈ W q,1 (γ ), q = p/( p−1). (4.6) X

X

If v = D H f , where f ∈ W p,2 (γ ), then δv = L f .

25 Sobolev classes on infinite-dimensional spaces

The function δv is called the divergence of the vector Veld v. It plays an important role in stochastic analysis. For a constant vector Veld v = h ∈ H we have δv =  h. For vector Velds v on R∞ of the simplest form v(x) = u(x1 , . . . , xn )ek the divergence is also easily evaluated: δv(x) = (D H u(x1 , . . . , xn ), ek ) H + u(x1 , . . . , xn )xk . Similarly, in a slightly more general case of the Veld v = u1h1 + · · · + un hn , we have δv =

u i ∈ W p,1 (γ ), h i ∈ H,

n    (D H u i , h i ) H + u i hi . i=1

The previous theorem says essentially that the L p -norm of this function can be controlled through the Sobolev norm of v; this is quite easy for p = 2, but requires some work in the general case. In fact, the following result is true (see, e.g., Shigekawa [83, Theorem 4.17] for the proof). Theorem 4.11. The divergence operator δ extends to a continuous linear operator δ : W p,r+1 (γ , H ) → W p,r (γ ). An analogous result is true for mappings with values in the space of Hilbert–Schmidt operators between H and a separable Hilbert space E, in which case the divergence takes values in E.

5 Inequalities and embeddings In the theory of Sobolev classes on Rn , a very important role is played by various inequalities related to embeddings of Sobolev classes into other functional classes such as L p -spaces. For example, every function f of class W 1,1 (Rn ) belongs not only to L 1 (Rn ), but also to L n/(n−1) (Rn ). There are also local embeddings of this sort. For example, any function in W p,1 (Rn ) with p > n has a continuous version. The Sobolev classes we discuss are analogs of weighted Sobolev classes on Rn . For the latter even on the real line the usual global embeddings fail. For example, the 2 function f (x) = ϕ(x)e−x e x /4 , where ϕ is smooth, ϕ(x) = 0 if x ≤ 0 and ϕ(x) = 1 if x ≥ 1, belongs to all classes W 2,r (γ ) for the standard Gaussian measure on the real line, but is in no class L p (γ ) with p > 2.

26 Vladimir I. Bogachev

Nevertheless, local embeddings hold for reasonable weights (say, positive continuous). In the inVnite-dimensional case, there are no even local embeddings. In particular, no continuity is ensured by the membership even in all the simplest example of a W p,r (γ ), p < ∞, r ∈ N. This is seen  in −1 n xn on R∞ with the stanmeasurable linear functional f (x) = ∞ n=1 dard Gaussian measure γ . Indeed, let us show that there is no function g continuous on R∞ and equal f a.e. (the fact that f itself is not continuous, is obvious, since every continuous linear function on R∞ depends on Vnitely many variables). Otherwise there is a neighborhood of zero V such that | f (x)| ≤ M a.e. in V for some M. Hence there exist k and c such that | f (x)| ≤ M a.e. on the  set S = {|xi | < c, i = 1, . . . , k}. There −1  n x is N such that  ∞ n ≤ N with probability at least 1/2. Since n=k+2 the set {x : xk+1 > ck + N + M + 1} has positive measure, we arrive at the contradiction: there is a positive measure set of points x ∈ S such that f (x) > M. Moreover, by using the previous example, one can Vnd a function f with compact support in R∞ belonging to all classes W p,r (γ ) for the standard Gaussian measure and having no continuous version. Also a function f with compact support can be constructed such that f ∈ W 2,k (γ ) p (γ ) with p > 2. Such for all k, but f does not belong to the union of L a function can be constructed in the form f = ∞ n=1 f n , where f n is a function of xn of class C0∞ . Nevertheless, there are some dimension-free inequalities that extend to the inVnite-dimensional case. Two important inequalities central for Gaussian analysis are presented in the next theorem. Theorem 5.1. Suppose that γ is a centered Radon Gaussian measure on a locally convex space X. Then, for any f ∈ W 2,1 (γ ), one has the logarithmic Sobolev inequality       1 2 2 2 2 f log | f | dγ ≤ |D H f | H dγ + f dγ log f dγ . (5.1) 2 X X X X In addition, there holds the Poincar´e inequality  

2

 f −

X

f dγ X

 dγ ≤ X

|D H f |2H dγ .

Moreover, if p ≥ 1, then p       p p  f − f dγ  dγ ≤ (π/2) M p |D H f | H dγ ,  X

X

X

(5.2)

(5.3)

27 Sobolev classes on infinite-dimensional spaces

where M p is the moment of order p of the standard Gaussian measure on the real line. Several authors contributed in discovering these inequalities in different form; Nash’s paper [68] is the earliest one I know where the Poincar´e inequality is explicitly given in the stated form with gradients (certainly, when written in terms of the Hermite expansions it becomes trivial); Stam [86] proved the logarithmic Sobolev inequality, later different derivations via hypercontractivity were found, see Federbush [43] and Nelson [69]; the paper of Gross [52] (where (5.1) was proved explicitly with gradients) became a starting point of intensive research related to characterizations and applications of logarithmic Sobolev inequalities (see references in Blanchere et al. [11], Bogachev [16], Lieb, Loss [64], and also the recent paper Cianchi, Pick [31]). As an application of the logarithmic Sobolev inequality let us consider the following situation that often arises in stochastic analysis. Suppose that ν =  · γ is a probability measure, where γ is the standard Gaussian measure on Rn or on R∞ . Its entropy (or the entropy of ) is deVned by  Entγ  :=  log  dγ , whenever  log  is integrable; otherwise we set Entγ  := +∞. Since the function t → t log t is convex and the logarithm of the integral of  vanishes, we have by Jensen’s inequality that Entγ  ≥ 0. Upper bounds for entropy are often of interest in applications. Suppose √ that we have  ∈ W 2,1 (γ ). Then the Sobolev inequality yields the estimate  |D H |2 1 Entγ  ≤ I(), I() := dγ , 2  where |D H |2 / = 0 on the set { = 0} and ∇ is the . To justify this we consider the standard Gaussian measure on Rn and √ note that  ∈ W 1,1 (γ ) and that ∇  = 2−1 −1/2 ∇ with the above convention. Indeed, the integrability of  and |∇|2 / yield the integrability of |∇| by the Cauchy inequality. We can calculate the derivatives pointwise by using the corresponding versions. The logarithmic Sobolev inequality is a certain weak replacement for missing analogs of the classical Sobolev inequalities in Rd which improve the initial integrability of a function on the basis of the integrability

28 Vladimir I. Bogachev

of its derivative. This is used, e.g., in the study of invariant measures of inVnite-dimensional diffusions (see Bogachev, R¨ockner [23]). The logarithmic Sobolev inequality is known to be equivalent to the so-called hypercontractivity of the Ornstein–Uhlenbeck semigroup. Theorem 5.2. The Ornstein–Uhlenbeck semigroup {Tt }t≥0 is hypercontractive, i.e., whenever p > 1, q > 1, one has Tt f q ≤  f  p for all t > 0 such that e2t ≥ (q − 1)/( p − 1). Applying this theorem we obtain a number of important results for polynomials. Corollary 5.3. Let p ≥ 2. Then the operator In : f → In ( f ) from L 2 (γ ) to L p (γ ) is continuous and In ( f ) p ≤ ( p − 1)n/2  f 2 .

(5.4)

In addition, for every p ∈ (1, ∞), the operators In are continuous on L p (γ ) and (5.5) In L(L p (γ )) ≤ (M − 1)n/2 ,   −1 where M = max p, p( p − 1) . Corollary 5.4. Let f ∈ Xd . For any α ∈ (0, d/(2e)), there holds the inequality

γ x : | f (x)| ≥ t f 2 ≤ c(α, d) exp(−αt 2/d ), d . d − 2eα Corollary 5.5. The spaces Xd are closed with respect to convergence in d  Xk that converges in measure, measure. Moreover, any sequence from

where c(α, d) = exp α +

k=0

is convergent in L p (γ ) for every p ∈ [1, ∞). The same is true for the spaces Xd (E) of mappings with values in any separable Hilbert space E. Corollary 5.6. The norms from L p (γ ), p ∈ [1, ∞), are equivalent on every Xn . In addition, for every p > 0, the topology on Xn induced by the metric from L p (γ ) coincides with the topology of convergence in measure. Finally, if q > p > 1, one has   q − 1 n/2 (5.6)  f  p ∀ f ∈ Xn .  f  p ≤  f q ≤ p−1

29 Sobolev classes on infinite-dimensional spaces

It should be noted that the classes W p,r (γ ) can be also deVned by completing the set of measurable polynomials with the respect to the Sobolev norm. Embedding inequalities in the case p = 1 are studied in Shigekawa [82]. Riesz transforms and other operators related to Gaussian L p - and Sobolev spaces are studied in Aimar, Forzani, Scotto [2], Brandolini, Chiacchio, Trombetti [26], Sj¨ogren, Soria [84]. Besov-type Gaussian spaces, a recent topic in this area, are studied in Pineda, Urbina [76] and Nikitin [70–72].

6 Sobolev classes over differentiable measures We now turn to general Fomin differentiable measures. Let μ be a Radon probability measure on a locally convex space X and let H be a separable Hilbert space continuously and densely embedded into X; as above, one can think that X = R∞ and H = l 2 . Sobolev classes over μ can be introduced in the same three ways as in the Gaussian case: as completions, via integration by parts, and using semigroups. We only give these deVnitions, referring the reader to Chapter 8 of the book [16], where a detailed discussion with proofs is given. Let {en } be an orthonormal basis in H . For p ≥ 1 and r ∈ N the Sobolev norm  ·  p,r is deVned by the formula 1/ p r    ∞  2  p/2  μ(dx) . (6.1) ∂ei1 · · · ∂eik f  f  p,r := k=0

X i 1 ,...,i k =1

The same norm can be written as r  D kH f  L p (μ,Hk ) .  f  p,r = k=0

For example, for r = 1 we obtain  f  p,1 =  f  L p (μ) + D H f  L p (μ,H ) . If f ∈ FC ∞ , then  f  p,r < ∞ since D kH f Hk ∈ Cb (X). Denote by p,r W p,r (μ) or by W H (μ) the completion of FC ∞ with respect to the norm  ·  p,r . The next deVnition introduces Sobolev functions by means of versions possessing directional derivatives with suitable integrability conditions. Let E be one more separable Hilbert space. DeHnition 6.1. Let F : X → E be μ-measurable. The mapping F is called absolutely ray continuous if, for every h ∈ H , the mapping F has a modiVcation Fh such that for every x ∈ X the mapping t → Fh (x + th) is absolutely continuous on bounded intervals.

30 Vladimir I. Bogachev

DeHnition 6.2. A μ-measurable mapping F : X → E is stochastically Gˆateaux differentiable if there exists a measurable mapping D H F : X → H(H, E) such that for every h ∈ H we have F(x + th) − F(x) − DH F(x)(h) −−→ 0 in measure μ. t→0 t The derivative of the n th order D nH F is deVned inductively as D H (D n−1 H F). An alternative notation is ∇ Hn F. DeHnition 6.3. Let 1 ≤ p < ∞. The space D p,1 (μ, E) is deVned as the continuclass of all mappings f ∈ L p (μ, E) such that f is ray absolutely   ous, stochastically Gˆateaux differentiable and D H f ∈ L p μ, H(H, E) . p,1 The space D p,1 (μ, E) (another notation is D H (μ, E)) is equipped with the norm  f 0p,1,E :=  f  L p (μ) + D H f  L p (μ,H(H,E)) . p,r

For r = 2, 3, . . . we deVne the classes D p,r (μ, E) = D H (μ, E) inductively:

  D p,r (μ, E) := f ∈ D p,r−1 (μ, E) : D H f ∈ D p,r−1 μ, H(H, E) . The corresponding norms are deVned by the equalities  f 0p,r,E :=  f  L p (μ) +D H f  L p (μ,H(H,E)) +· · ·+DrH f  L p (μ,Hr (H,E)) . p,r

We set D p,r (μ) := D H (μ) := D p,r (μ, R). Let us turn to the deVnition with generalized derivatives. Let j : X ∗ → H be the adjoint embedding for the embedding H → X, i.e., we have ( j (l), h) H = l(h) for all l ∈ X ∗ , h ∈ H . Since H is dense in X, we see that j (X ∗ ) is dense in H . DeHnition 6.4. We shall say that a function f ∈ L1 (μ) has a generalized partial derivative g = ∂h f ∈ L 1 (μ) along a vector h ∈ H if fβhμ ∈ L1 (μ) and for every ϕ ∈ FC ∞ one has   ∂h ϕ(x) f (x) μ(dx) = − g(x)ϕ(x) μ(dx) X X (6.2) μ − f (x)ϕ(x)βh (x) μ(dx). X

31 Sobolev classes on infinite-dimensional spaces

It is clear that a generalized partial derivative is uniquely determined as an element of L 1 (μ). p,1

DeHnition 6.5. Let G p,1 (μ) = G H (μ) be the class of all real functions f ∈ L p (μ) possessing generalized partial derivatives along all vectors in j (X ∗ ) and having Vnite norms  f  p,1 :=  f  L p (μ) + D H f  L p (μ,H ) < ∞, where D H f is deVned as follows: there is a mapping T : X → H such that for every l ∈ X ∗ we have l, T (x) = ∂ j (l) f (x) a.e. Then we set D H f := ∇ H f := T . The space G p,1 (μ) is equipped with the norm p,1  ·  p,1 . Similarly we deVne the class G p,1 (μ, E) = G H (μ, E) of mappings with values in a Hilbert space E. Hence one can inductively p,r introduce the classes G p,r (μ) = G H (μ) of functions f ∈ L p (μ) with D kH f ∈ L p (μ, Hk ) whenever k ≤ r equipped with the norms  f  p,r :=  f  L p (μ) +

r 

D kH f  L p (μ,Hk ) .

k=1

In our model example X = R∞ and H = l 2 , the inclusion f ∈ G p,1 (μ) that means that f ∈ L p (μ) has generalized partial derivatives ∞ ∂en f such 2 p 2 ∈ L (μ, H ), where |D f | = |∂ f | . CerD H f := (∂en f )∞ H e n H n=1 n=1 tainly, the inclusions fβ μj (l) ∈ L 1 (μ) for all l ∈ X ∗ are implicitly meant (here this reduces to fβeμn ∈ L 1 (μ) for all n). Proposition 6.6. Let p ≥ 1 and βhμ ∈ L p/( p−1) (μ) for all h ∈ j (X ∗ ). Then the spaces G p,1 (μ, E) with the respective norms are complete. Remark 6.7. Let p ≥ 1 and βhμ ∈ L p/( p−1) (μ) for all h ∈ H . If f ∈ G p,1 (μ), then f has generalized partial derivatives ∂h f ∈ L p (μ) for all h ∈ H , not only for the elements of j (X ∗ ), as required by the deVnition. Indeed, let {en } be an orthonormal basis in H contained in j (X ∗ ); in the case of R∞ just the usual basis. Since βhμ ∈ L p/( p−1) (μ), we obtain a linear mapping H → L p/( p−1) (μ). n It is readily seen that its graph is closed. Therefore, letting h n = i=1 (h, ei ) H ei , we obtain convergence βhμn → βhμ in L p/( p−1) (μ). Therefore, (6.2) remains valid for h once it holds for each h n . Finally, there is an exact analog of the deVnition involving a symmetric semigroup (which has been used to obtain fractional classes H p,r in the Gaussian case). Let {en } be an orthonormal basis in H such that en = j (ln ), ln ∈ X ∗ ; it is again wise to assume that we deal with the space X = R∞ and H = l 2 ,

32 Vladimir I. Bogachev

in which case we take the standard basis. Suppose that βeμn ∈ L 2 (μ) ∀ n ∈ N. Then we obtain the operator Lf =

n  [∂e2i f + βeμi ∂ei f ] i=1

acting on functions f of the form f (x) = f 0 (l1 (x), . . . , ln (x)), f 0 ∈ Cb∞ (Rn ); that is, on smooth cylindrical functions in the case of R∞ . This operator is densely deVned and symmetric in L 2 (μ):  n   g(x)L f (x) μ(dx) = − ∂ei g(x)∂ei f (x) μ(dx) X

X

i=1 = − (D H f, D H g) H dμ X

due to the integration by parts formula. In addition, (L f, f )2 ≤ 0. Therefore, the operator L has a non-positive selfadjoint extension, namely, we take its Friedrichs extension denoted by the same symbol L (see Reed, Simon [80, Section X.3]). The domain of deVnition of this extension will be denoted by D(L). The bounded operator (I − L)−1 is a selfadjoint nonnegative contraction on L 2 (μ). Hence we can also deVne the powers (I − L)−r/2 on L 2 (μ) and obtain the Sobolev spaces   H 2,r (μ) := (I − L)−r/2 L 2 (μ) , r > 0. −r

However, we can get even more: one can show that the operators (I − L) 2 extend as nonnegative contractions on the spaces L p (μ), 1 ≤ p < ∞, so that the Sobolev classes   H p,r (μ) := (I − L)−r/2 L p (μ) , r > 0, p ≥ 1,

arise. Unlike the Gaussian case, their relation (for natural values of r) to the previously deVned classes has not been clariVed. Moreover, the exact relations between the other classes remain unclear except for rather special cases. One of them is considered in part (iii) of the next theorem. Theorem 6.8. (i) If βhμ ∈ L p/( p−1) (μ) for all h ∈ j (X ∗ ), then we have W p,r (μ) ⊂ G p,r (μ) and for all p ∈ [1, ∞), r ∈ N.

D p,r (μ) ⊂ G p,r (μ)

33 Sobolev classes on infinite-dimensional spaces

(ii) Suppose that for every h ∈ j (X ∗ ) the corresponding conditional measures μ y on the real line have continuous positive  densities (for example, this holds if there exist ch > 0 such that exp ch βhμ ∈ L 1 (μ)). Then D p,1 (μ) = G p,1 (μ). (iii) Suppose that X = R∞ , condition (ii) is fulFlled and that n n      βeμi ei − IEn βeμi ei  sup n

i=1

i=1

L p (μ,l 2 )

< ∞,

(6.3)

where IEn is the conditional expectation with respect to the σ -algebra generated by the coordinates x1 , . . . , xn . Then W p,1 (μ) = D p,1 (μ) = G p,1 (μ) for all p > 1. Condition (6.3) here is very restrictive (but it holds for product-measures and some measures absolutely continuous with respect to product-measures). It would be interesting to Vnd more general conditions ensuring the coincidence of the classes W p,1 (μ), D p,1 (μ), and G p,1 (μ); also their relation to H p,1 (μ) is open in the general case. The interpolation approach to fractional Sobolev classes was suggested in Watanabe [89] and further employed by several authors, see, e.g., Airault, Bogachev, Lescot [3], Bogachev [16], Nikitin [70–72].

7 The class BV: the Gaussian case In a particular way one introduces the space BV (γ ) of functions of bounded variation, containing W 1,1 (γ ), see Fukushima [49], Fukushima, Hino [50], Hino [55–57], Ambrosio, Miranda, Maniglia, Pallara [8, 9], Bogachev, Pilipenko, Rebrova [20], Bogachev, Rebrova [22], Bogachev, Shaposhnikov [24], and R¨ockner, Zhu, Zhu [81]. In the case of R∞ with the standard Gaussian measure it consists of functions f ∈ L 1 (γ ) such that xn f ∈ L 1 (γ ) for all n and there is an H -valued measure f of bounded variation for which the scalar measures ( f, en ) H satisfy the identity   ϕ(x) ( f, en ) H (dx) = − [ f (x)∂en ϕ(x) − xn f (x)ϕ(x)] γ (dx) for all ϕ ∈ FCb∞ . In the general case the deVnition is the same, we just en in place take for {en } an orthonormal basis in H and use the functionals of the coordinate functions xn . So the general deVnition reads as follows.

34 Vladimir I. Bogachev

DeHnition 7.1. The class BV (γ ) consists of all functions f ∈ L 1 (γ ) such that f  h ∈ L 1 (γ ) for all h ∈ H and there is an H -valued measure f of bounded variation satisfying the identity   ϕ(x) ( f, h) H (dx) = − [ f (x)∂h ϕ(x) −  h(x) f (x)ϕ(x)] γ (dx) (7.1) X

for all ϕ ∈ FC ∞ and all h ∈ H . If f ∈ W 1,1 (γ ), then we take f = ∇ f · γ and see that f ∈ BV (γ ), since xi f ∈ L 1 (γ ) by Lemma 4.5. The following fact is known (see, e.g., Ambrosio, Miranda, Maniglia, Pallara [9]). Theorem 7.2. A function f ∈ L 1 (γ ) belongs to BV (γ ) precisely when sup Tt f 1,1 < ∞. t>0

This is also equivalent to the existence of a sequence of functions f n ∈ FC ∞ convergent to f in L 1 (γ ) and bounded in W 1,1 (γ ). We now consider a broader class of functions of bounded variation: its deVnition is based on vector measures of semibounded variation in place of measures of bounded variation. DeHnition 7.3. The class S BV (γ ) consists of all functions f ∈ L 1 (γ ) such that f  h ∈ L 1 (γ ) for all h ∈ H and there is an H -valued measure f of semibounded variation satisfying (7.1). In the Vnite-dimensional case the classes BV (γ ) and S BV (γ ) coincide as sets, but their norms are different. Already for smooth functions f , where f is given by a vector density ∇ f with respect to γ , the BV norm may be much larger, since it involves the integral of |∇ f |, while the S BV -norm deals with the integrals of |∂h f | with |h| ≤ 1 (see the example below where this is shown explicitly). Lemma 7.4. There is a number C > 0 such that    h f  L 1 (γ ) ≤ C  f 1 + V ( f ) , f ∈ S BV (γ ), |h| H ≤ 1. Proof. This can be derived by using Lemma 4.5 (in fact, by Theorem 7.2, it sufVces to consider f ∈ W 1,1 (γ )) or by a similar reasoning for functions of class S BV . Proposition 7.5. The space BV (γ ) is Banach with the norm  f  BV =  f 1 +  f .

35 Sobolev classes on infinite-dimensional spaces

The space S BV (γ ) is Banach with the norm  f  S BV =  f 1 + V ( f ). Proof. Let { f n } be a Cauchy sequence in BV (γ ). Then it converges in L 1 (γ ) to some function f ∈ L 1 (γ ). In addition, the measures f n converge in variation to some H -valued measure . Finally, for each h ∈ H the sequence of functions  h f n is fundamental in L 1 (γ ) by the estimate h f . Therefore, we in the lemma above, hence it converges in L 1 (γ ) to  can set f := and obtain (7.1), i.e., f ∈ BV (γ ). By construction, f n → f in BV (γ ). The proof of the second assertion is the same. Example 7.6. In the inVnite-dimensional case BV (γ )  = S BV (γ ). For the proof we have to Vnd a function f ∈ S BV (γ ) for which the variation of f is inVnite. It sufVces to verify that on every Rk we can Vnd a smooth function f k such that the ratio of the variation of the measure f k and its semivariation tends to inVnity as k → ∞. As indicated above, f k = ∇ f k · γk , where γk is the standard Gaussian measure on Rk . In addition,  f k  is the integral of |∇ f k | with respect to the measure γk , V ( f k ) is the maximum of the integrals of |∂h f k | with respect to the measure γk taken over h in the unit ball. Let us take f k of the form g(|x|2 ), where g is a smooth nonzero function with support in [0, 1]. Then V ( f k ) is the integral of |∂x1 f k | with respect to the measure γk . Passing to spherical coordinates we obtain that V ( f k )/ f k  is the ratio of the integrals of cos ϕ sink−2 ϕ and sink−2 ϕ over [0, π/2]. It is straightforward to verify that this ratio tends to zero, since the integral of the function sink−2 ϕ over [0, tk ], where tk = arccos k −1/4 , is estimated by C exp(−k 1/2 /2), the integral of sink−2 ϕ over [0, π/2] is greater than the integral of cos ϕ sink−2 ϕ, which equals (k − 1)−1 , and on [tk , π/2] one has cos ϕ ≤ k −1/4 .

8 The class BV: the general case In this section we consider BV -functions over general Fomin differentiable measures. Suppose that μ is a Radon probability measure on a locally convex space X that is Fomin differentiable (i.e., has a logarithmic derivative) along all vectors from a continuously embedded Hilbert space H . For simplicity, one can assume that X = R∞ and H = l 2 . As in the Gaussian case, there are two options in the deVnition of functions of bounded variation: based on vector measures of bounded and semibounded variation. The proofs of the results of this section can be found in Bogachev, Rebrova [22] or in the more general case of domains

36 Vladimir I. Bogachev

in the last section and in the papers Bogachev, Pilipenko, Rebrova [20], Bogachev, Pilipenko, Shaposhnikov [21]. Recall that βhμ is the logarithmic derivative of μ along h. For the subsequent discussion the following facts may be useful. For any measure μ differentiable along H we obtain an H -valued measure Dμ deVned by the equality (Dμ(B), h) H = dh μ(B). If H is inVnite-dimensional and μ is not zero, then the measure Dμ has unbounded variation (see Proposition 7.3.2 in Bogachev [16]). However, if Dμ is regarded as an X-valued measure and X is a Banach space, then under broad assumptions this X-valued measure has bounded variation (e.g., if the embedding H → X is absolutely summing, see Chapter 7 in Bogachev [15]). If μ is a centered Gaussian measure and H is its Cameron–Martin space, then Dμ as an X-valued measure has vector density −x with respect to μ. Note that the vector measure with density n −x  with respect to the standard Gaussian measure on R has variation |x| γ (x), which goes to +∞ as n → +∞, but its semivariation is independent of n. Set M H (μ) = { f ∈ L 1 (μ) : fβhμ ∈ L 1 (μ) ∀ h ∈ H }. Theorem 8.1. The set M H (μ) is a Banach space with the norm  f  M :=  f  L 1 (μ) + sup  fβhμ  L 1 (μ) . |h| H ≤1

Proof. For each f ∈ M H (μ) the quantity  f  M is Vnite, since the mapping h → fβhμ from H to L 1 (μ) has a closed graph. Indeed, if h n → h in H and fβhμn → g in L 1 (μ), then g = fβhμ , because by the continuity of the embedding H → D D(μ) (which follows by the closed graph theorem) we have βhμn → βhμ in L 1 (μ), hence in measure. Thus, the operator h → fβhμ is bounded, so  f  M < ∞. Now, if { f n } is a Cauchy sequence in this norm, it converges to a function f in L 1 (μ). Clearly,  f  M < ∞. Moreover, given ε > 0, we take N such that  f n − f k  M ≤ ε for all n, k ≥ N and by Fatou’s theorem conclude that  f n − f  M ≤ ε for all n ≥ N. DeHnition 8.2. Let SV (μ) = { f ∈ L 1 (μ) : the Skorohod derivative dh ( f · μ) exists for all h ∈ H }, S BV (μ) = SV (μ) ∩ M H (μ).

37 Sobolev classes on infinite-dimensional spaces

In other words, the class S BV (μ) consists of all functions f ∈ L 1 (μ) for which sup | fβh | L 1 (μ) < ∞ |h|≤1

and there exists an H -valued measure f of bounded semivariation such that the Skorohod derivative dh ( f μ) exists and equals ( f, h) H + fβh μ for each h ∈ H . It is important to note that the measure ( f, h) H can be singular with respect to μ (say, have atoms in the one-dimensional case), but it also admits a disintegration y,h

( f, h) H = ( f, h) H μY (dy) y,h

with some measures ( f, h) H on the straight lines y + Rh, where y ∈ Y and Y is a closed hyperplane complementing Rh. Indeed, we have ( f, h) H = dh ( f μ) − fβh μ, where the projections of |dh ( f μ)| and | fβh |μ on Y are absolutely continuous with respect to the projection of μ, since the projection of the measure |dh ( f μ)| is absolutely continuous with respect to the projection of | f |μ. The latter follows from the fact that the projection on Y of the Skorohod derivative dh σ of a nonnegative measure σ on X is absolutely continuous with respect to the projection of σ (although dh σ itself need not be absolutely continuous with respect to σ unlike the case of Fomin’s derivative), because for any Borel set B ⊂ Y with σY (B) = 0 we have  dh σ (Rh × B) = dh σ y (Rh × B) σY (dy) Y = dh σ y (Rh × B) σY (dy) = 0. B

Theorem 8.3. (i) The set SV (μ) is a Banach space with the norm  f  SV :=  f  L 1 (μ) + sup dh ( f · μ), |h| H ≤1

and for every function f ∈ SV (μ) there is an H -valued measure D( f ·μ) of bounded semivariation such that dh ( f · μ) = (D( f · μ), h) H for all h ∈ H. (ii) The set S BV (μ) is a Banach space with the norm  f  S BV :=  f  M +  f  SV , and for every function f ∈ S BV (μ) there is an H -valued measure f of bounded semivariation such that dh ( f · μ) = ( f, h) H + f · dh μ for all h ∈ H .

38 Vladimir I. Bogachev

DeHnition 8.4. Let BV (μ) be the class of all functions f ∈ S BV (μ) such that the H -valued measure f has bounded variation. Theorem 8.5. The set BV (μ) is a Banach space with the norm  f  BV :=  f  S BV +  f . For an interval J ⊂ R (possibly unbounded) let BVloc (J ) denote the class of all functions on J having bounded variation on every compact interval in J . Lemma 8.6. A function f ∈ M H (μ) belongs to S BV (μ) precisely when there is an H -valued measure f of bounded semivariation such that for every h ∈ H for μ-almost every x the function t → f (x + th) belongs to BVloc (R) and its generalized derivative is x,h ( f, h)x,h (t), H /

where  x,h is the density of the conditional measure μx,h for μ on the straight line x + Rh. A similar assertion is true for BV (μ), where the measure must have bounded variation. Example 8.7. Let X = H = Rn , let μ be a probability measure with a smooth density , and let f be a smooth function such that f , |∇ f | and f |∇|/ are integrable with respect to μ. Then dh μ is a measure with density ∂h , dh ( f · μ) is a measure with density ∂h ( f ) = f ∂h  + ∂h f , whence it follows that f is a measure with vector density (∇ f ) with respect to Lebesgue measure, i.e., with density ∇ f with respect to the measure μ. If the function f ∈ L 1 (μ) with f |∇| ∈ L 1 (Rn ) is not smooth, but belongs to the class BVloc (Rn ) of locally integrable functions whose generalized Vrst order derivatives are locally bounded measures, then f belongs to BV (μ) and f =  · D f provided that the latter measure is bounded. It is clear that in the Vnite-dimensional case the classes S BV and BV coincide as sets and their norms are equivalent, but these norms are different. Theorem 8.8. The spaces BV (μ) and S BV (μ) possess the following property: if a sequence of functions f n is norm bounded in it and converges almost everywhere to a function f , then f belongs to the same class and the norm of f does not exceed the precise upper bound of the norms of the functions f n . Functions of bounded variation on spaces with convex measures are considered in Ambrosio, Da Prato, Goldys, Pallara [4].

39 Sobolev classes on infinite-dimensional spaces

9 Sobolev functions on domains and their extensions Let us proceed to domains. In this section, we introduce Sobolev classes on domains, and BV -functions will be discussed in the next section. Given a set U ⊂ X, the symbol L p (U, μ) will denote the space of equivalence classes of all μ-measurable functions f on U for which the functions | f | p are integrable with respect to the measure μ on U . We Vrst consider the Gaussian case and then brieWy discuss the case of a general differentiable measure. Let V ⊂ X. If the sets (V − x) ∩ H are open in H for all x ∈ V , then V is called H -open (this property is equivalent to the fact that V − x contains a ball from H for every x ∈ V , and is weaker than openness of V in X), and if all such sets are convex, then V is called H -convex. The latter property is weaker than the usual convexity. Obviously, for any H -convex and H -open set V all nonempty sets Vx,h are open intervals (possibly unbounded), where Vx,h := V ∩ (x + Rh). Example 9.1. Any open convex set is H -open and H -convex. However, the convex ellipsoid ∞ 

 n −2 xn2 < 1 U = x ∈ R∞ : n=1

is not open in R∞ , but is H -open, where H = l 2 . This ellipsoid has positive measure with respect to the standard Gaussian product-measure γ . The set N 

 xn2 = 1 Z = x ∈ R∞ : lim N −1 N →∞

n=1

is Borel and has full measure with respect to γ (by the law of large numbers). It is not convex: if x ∈ Z , then −x ∈ Z , but 0  ∈ Z . It is clear that Z has no interior (its intersection with the set of Vnite sequences is empty). However, Z is H -open and H -convex, since for every x ∈ Z we have (Z − x) ∩ H = H , that is, for every h ∈ H we have x + h ∈ Z . Indeed, N −1

N N N    (xn + h n )2 − N −1 xn2 = N −1 (h 2n + 2xn h n ), n=1

n=1

n=1

N h 2n → 0 for each which tends to zero as N → ∞, because N −1 n=1  N −1 h ∈ H and |N n=1 x n h n | → 0 by the Cauchy inequality.

40 Vladimir I. Bogachev

Suppose now that we are given a Borel or γ -measurable set V ⊂ X of positive γ -measure such that its intersection with every straight line of the form x + R1 en is a convex set Vx,en . There are several natural ways of introducing Sobolev classes on V . The Vrst one is considering the class W p,1 (V, γ ) equal to the completion of FC ∞ with respect to the Sobolev norm  ·  p,1,V with the order of integrability p, evaluated with respect to the restriction of γ to V . This class is contained in the class D p,1 (V, γ ) consisting of all functions f on V belonging to L p (V, γ ) and having versions of the type indicated above, but with the difference that now the absolute continuity is required only on the closed intervals belonging to the sections Vx,en . The class D p,1 (V, γ ) is naturally equipped with the Sobolev norm  ·  p,1,V deVned by the restriction of γ to V : 

1/ p  1/ p p | f | dγ + |∇ f | dγ .

 f  p,1,V =

p

V

V

In the paper Hino [57] the Sobolev class D 2,1 (V, γ ) was used (denoted there by W 1,2 (V )). In the Vnite-dimensional case for convex V both classes coincide, the inVnite-dimensional situation is less studied, but for H -convex H -open sets one has W 2,1 (V, γ ) = D 2,1 (V, γ ), which follows from [57], where it is shown that D 2,1 (V, γ ) contains a dense set of functions possessing extensions of class W 2,1 (γ ) and for this reason belonging to W 2,1 (V, γ ). It is readily veriVed that the spaces W p,1 (V, γ ) and D p,1 (V, γ ) with the Sobolev norm are Banach. Note that one can introduce more narrow Sobolev classes on V that admit extensions. For example, in the space W p,1 (V, γ ) one can take the p,1 closure W0 (V, γ ) of the set of functions from W p,1 (γ ) with compact p,1 support in V ; the functions from W0 (V, γ ) extended by zero outside p,1 of V belong to W (γ ). For certain very simple sets V , say, half-spaces, it is easy to deVne explicitly an extension operator. Example 9.2. Let V = {x :  h(x) > 0}, where |h| H = 1. Then any function f ∈ W p,1 (V, γ ) has an extension of class W p,1 (γ ) deVned by  f (x) = f (x − 2 h(x)h) if  h(x) < 0. For example, if X = R∞ and h = e1 , then V = {x1 > 0} and  f (x1 , x2 , . . .) = f (−x1 , x2 , . . .) whenever x1 < 0.

41 Sobolev classes on infinite-dimensional spaces

Taking a sequence of functions f j ∈ FC ∞ whose restrictions to V confj verge to f in the norm of W p,1 (γ , V ), we see that the functions  redeVned on the set { h ≤ 0} by  f j (x) = f j (x − 2 h(x)h) belong to f j  L p (γ ) = W p,1 (γ ) and converge in the Sobolev norm. In particular,   1/ p 1/ p  p p p 2  f j  L (V,γ ) , D H f j  L (γ ) = 2 D H f j  L (V,γ ) . It is not clear whether there are essentially inVnite-dimensional domains V for which all Sobolev functions have extensions. However, functions with bounded derivatives extend from any H -convex domains. Proposition 9.3. Let V be H -convex and let f ∈ W 2,1 (γ , V ) be such that |D H f (x)| ≤ C a.e. in V . Then there is a function g ∈ W p,1 (γ ) for all p < ∞ such that g|V = f |V a.e. on V and |g(x + h) − g(x)| ≤ C|h| H for all x ∈ X and h ∈ H . Lemma 9.4. If the set V is such that for some p ≥ 1 every function f ∈ W p,1 (V, γ ) has an extension g ∈ W p,1 (γ ), then there exists an extension g f ∈ W p,1 (γ ) such that g f  p,1 ≤ C f  p,1,V with some common constant C. The following theorem is a negative result about extensions. For notational simplicity, we formulate it for the Gaussian product-measure on R∞ . A complete proof is given in Bogachev, Pilipenko, Shaposhnikov [21]. Theorem 9.5. The space R∞ contains a convex Borel H -open set K of positive γ -measure with the following property: for every p ∈ [1, +∞) there is a function in the class W p,1 (K , γ ) having no extensions to a function of class W p,1 (γ ). One can also Fnd a convex compact set K with the same property. Remark 9.6. In the proof of this theorem in [21] a certain Hilbert space L of full measure is taken such that, passing to the restriction of the measure γ to L, we obtain a convex and open in L set K of positive measure, on which for every p ∈ [1, +∞) there is a function in the class W p,1 (K , γ ) without restrictions to a function in W p,1 (γ ). It is clear that the same example can be realized also on a larger weighted Hilbert space of sequences in which K will be precompact. Hence it is possible to combine H -openness of K with its relative compactness in a Hilbert space (clearly, our set K in R∞ is relatively compact). In addition, if we take for γ the classical Wiener measure on the space C[0, 1] or L 2 [0, 1] and embed this space into R∞ by means of the mapping x → n(x, en ) L 2 , where {en } is the orthonormal basis in L 2 [0, 1]

42 Vladimir I. Bogachev

formed by the eigenfunctions of the covariance operator of the Wiener measure, that is, en (t) = cn sin((πn − π/2)t), n ∈ N, with the eigenvalues λn = (πn − π/2)−2 , then the image of γ will coincide with the standard Gaussian product-measure and the space L mentioned above will coincide with the image of L 2 [0, 1] under the embedding, hence our convex set K will be open in the corresponding Hilbert space. For any centered Radon Gaussian measure γ on a locally convex space X with the inVnite-dimensional Cameron-Martin space H , the results presented above yield existence of an H -open convex Borel set V of positive γ -measure and, for every p ∈ [1, +∞), a function f in W p,1 (V, γ ) without extensions to functions in the class W p,1 (γ ). It would be interesting to construct an example of a function in the intersection of all W p,1 (V, γ ) without extensions of class W 1,1 (γ ). Apparently, there are bounded functions with such a property. Certainly, it is natural to ask about such examples on a ball in a Hilbert space. However, we have no such examples.

10 BV functions on domains and their extensions In this section we follow the paper Bogachev, Pilipenko, Shaposhnikov [21]. We assume below that the measure μ on a locally convex space X is Fomin differentiable along all vectors in a separable Hilbert space H continuously and densely embedded into X (again the model example is l 2 ⊂ R∞ ) and that for every Vxed h ∈ H the continuous versions of the conditional densities on the straight lines x + Rh are positive (a sufVcient condition for this is the integrability of exp |βhμ | with respect to μ). Below these densities are denoted by  x,h without indicating μ. Let U ⊂ X be a Borel set that is H -convex and H -open, that is (see Section 9), all sets (U − x) ∩ H are convex and open in H . For example, this can be a set that is convex and open in X. For any H -convex and H -open set U the one-dimensional sections Ux,h := U ∩ (x + Rh), are open intervals on the straight lines x + Rh. We shall often identify these intervals with the intervals Jx,h := {t ∈ R : x + th ∈ U }. In particular, when speaking about functions on intervals Ux,h we shall mean sometimes functions of the real argument on Jx,h .

43 Sobolev classes on infinite-dimensional spaces

Let M H (U, μ) denote the class of all functions f ∈ L 1 (U, μ) such that   f  M :=  f  L 1 (U,μ) + sup | f (x)| |βh (x)| μ(dx) < ∞. |h|≤1 U

The corresponding space of equivalence classes will be denoted by the same symbol. This is the exact analog of the class M H (μ) in Section 8. Lemma 10.1. The set M H (U, μ) is a Banach space with the norm  f  M :=  f  L 1 (U,μ) + sup  fβhμ  L 1 (U,μ) . |h| H ≤1

Proof. Let us observe that the operator h → fβh from H to L 1 (U, μ) is linear and has a closed graph. Indeed, suppose that h n → h in H and fβh n → g in L 1 (U, μ). By the continuity of the embedding H → D(μ) we have βh n → βh in L 1 (μ), whence it follows that fβh n → fβh in measure on U , hence g = fβh . Therefore, for every f ∈ M H (U, μ) the quantity  f  M is Vnite. Obviously, it is a norm. Let { f n } be a Cauchy sequence in M H (U, μ). Then { f n } converges in L 1 (U, μ) to some function f . By Fatou’s theorem f ∈ M H (U, μ). In addition, f is a limit of { f n } with respect to the norm in M H (μ): if  f n − f k  M ≤ ε for all n, k ≥ n 1 , then  f n − f  M ≤ ε for all n ≥ n 1 . DeHnition 10.2. We shall say that f ∈ M H (U, μ) belongs to the class S BVH (U, μ) if the function t → f (x + th) x,h (t) belongs to the class BVloc (Ux,h ) for every Vxed h ∈ H for almost all x and there exists an H -valued measure U f on U of bounded semivariation such that, for every h ∈ H , the measure ( U f, h) H admits the representation ( U f, h) H = ( U f, h)x,h,μ μ(dx), H where the measures ( U f, h)x,h,μ on the straight lines x + Rh possess H the property that x,h ( U f, h)x,h (t) H + f (x + th)∂t 

is the generalized derivative of the function t → f (x + th) x,h (t) on Ux,h . The class BVH (U, μ) consists of all f ∈ S BVH (U, μ) such that the measure U f has bounded variation.

44 Vladimir I. Bogachev

In other words, an analog of the characterization from Lemma 8.6 is now taken as a deVnition. As we have warned above, the sections Ux,h in this deVnition are identiVed with intervals Jx,h of the real line. Note that the deVning relation for ( U f, h)x,h H can be stated in terms of the functions t → f (x + th) (not multiplied by conditional densities): the generalized derivatives of these functions must be (as in Lemma 8.6)  x,h (t)−1 ( U f, h)x,h H . An equivalent description of functions in S BVH (U, μ) can be given in the form of integration by parts if in place of the class FC ∞ we use appropriate classes of test functions for every h ∈ H . For any Vxed vector h ∈ H we choose a closed hyperplane Y complementing Rh and consider the class Dh of all bounded functions ϕ on X with the following properties: ϕ is measurable with respect to all Borel measures, for each y ∈ Y the function t → ϕ(y + th) is inVnitely differentiable and has compact support in the interval Jy,h = {t : y + th ∈ U }, and the functions ∂hn ϕ are bounded for all n ≥ 1. Here ∂hn ϕ(y + th) is the derivative of order n at the point t for the function t → ϕ(y + th). Note that ψϕ ∈ Dh for all ϕ ∈ Dh and ψ ∈ FC ∞ . Lemma 10.3. A function f ∈ M H (U, μ) belongs to S BVH (U, μ) precisely when there exists an H -valued measure U f on U of bounded semivariation such that, for every h ∈ H and all ϕ ∈ Dh , one has the equality   ∂h ϕ(x) f (x) μ(dx) = − ϕ(x) ( U f, h) H (dx) X X − ϕ(x) f (x)βh (x) μ(dx). X

A similar assertion with variation in place of semivariation is true for the class BVH (U, μ). Proof. If f ∈ S BVH (U, μ), then the indicated equality follows from the deVnition and the integration by parts formula for conditional measures. Let us prove the converse assertion. Let us Vx k ∈ N. The set Yk of all points y ∈ Y such that the length of the interval Jy,h is not less than 8/k is measurable with respect to every Radon measure (see Bogachev [15, Theorem 7.14.49]). In addition, it is not difVcult to show that there exists

45 Sobolev classes on infinite-dimensional spaces

a function gk ∈ Dh measurable with respect to every Borel measure and possessing the following properties: 0 ≤ gk ≤ 1, gk (y) = 0 if the length of Jy,h is less than 8/k, gk (y + th) = 0 if t ∈ Jy,h or if t ∈ Jy,h and the distance from t to an endpoint of Jy,h is less than 1/k, gk (y + th) = 1 if t ∈ Jy,h and the distance from t to an endpoint of Jy,h is not less than 2/k. It follows from our hypothesis that for all ψ ∈ FC ∞ we have the equality   ∂h ψ(x)gk (x) f (x) μ(dx) = − ψ(x)gk (x) ( U f, h) H (dx) X X − ψ(x)gk (x) f (x)βh (x) μ(dx) X − ψ(x)∂h gk (x) f (x) μ(dx). X

Therefore, the measure f gk μ is Skorohod differentiable and dh ( f gk μ) = gk ( U f, h) H + f gk βh μ + ∂h gk f μ. Using the disintegration for f gk μ and letting k → ∞, we obtain the disintegration for f μ required by the deVnition. Lemma 10.4. If f ∈ S BVH (U, μ) and ψ ∈ Cb1 (R), then ψ( f ) ∈ S BVH (U, μ) and for any h ∈ H one has ( U ψ( f ), h) H = ( U ψ( f ), h)x,h,μ μ(dx) H with

= ψ  ( f )(x + th)( U f, h), ( U ψ( f ), h)x,h,μ H

where ψ  ( f )(x + th) is redeFned at the points of jumps of the function t → f (x + th) by the expression ψ( f (x + th+)) − ψ( f (x + th−)) . f (x + th+) − f (x + th−) Moreover,

V ( U ψ( f )) ≤ L V ( U f ), supu=v |ψ(u)−ψ(v)| |u−v|

is the Lipschitz constant of ψ. where L = A similar assertion is true for BVH (U, μ). Proof. It sufVces to use conditional measures and apply the chain rule for BV-functions in the one-dimensional case, see, e.g., Ambrosio, Fusco, Pallara [6, page 188].

46 Vladimir I. Bogachev

Theorem 10.5. The set S BVH (U, μ) is a Banach space with the norm  f  S BV :=  f  M + V ( U f ). The set BVH (U, μ) is a Banach space with the norm  f  BV :=  f  M + Var( U f ). Proof. Since the space M H (U, μ) is complete, every Cauchy sequence { f n } in the space S BVH (U, μ) converges in the M-norm to some function f ∈ L 1 (U, μ). The sequence of measures U f n is Cauchy in semivariation, hence converges in the norm V to some measure ν of bounded semivariation. Applying Lemma 10.3 it is easy to show that f ∈ S BVH (U, μ) and ν = U f is the corresponding H -valued measure. Since  f n − f  M + V ( U f n − U f ) → 0, it follows that f is a limit of { f n } in the norm of the space S BVH (U, μ). The proof of completeness of the space BVH (U, μ) is similar. Theorem 10.6. The classes S BVH (U, μ) and BVH (U, μ) have the following property: if a sequence of functions { f n } is norm bounded in it and converges almost everywhere to a function f , then f belongs to the same class, and the norm of f does not exceed the precise upper bound of the norms of the functions f n . Moreover, for every Fxed h ∈ H , the measures ( U f n , h) H converge to ( U f, h) H in the weak topology generated by the duality with Dh (in the case U = X also with respect to the duality with FC ∞ ). Proof. These assertions are true on the real line, since our assumption about the conditional densities means that the density of μ is positive, so f n =  f n , where f n is the generalized derivative, and these measures converge to  f  in the sense of distributions. In the general case suppose Vrst that { f n } is uniformly bounded. Let us Vx h ∈ H . Since the measures ( U f n , h) H are uniformly bounded and μ(dx), ( U f n , h) H = ( U f n , h)x,h,μ H by Fatou’s theorem the function  lim inf ( U f n , h)x,h,μ H n→∞

47 Sobolev classes on infinite-dimensional spaces

is μ-integrable. In particular, it is Vnite almost everywhere, hence the restriction of the function f to almost every straight line x +Rh is in BVloc . on these straight lines Moreover, we obtain Vnite measures ( U f, h)x,h,μ H such that the measure μ(dx) ( U f, h) H := ( U f, h)x,h,μ H is Vnite for every h ∈ H . By the Pettis theorem (see Dunford, Schwartz [39, Chapter IV, §10]) we obtain an H -valued measure U f . It meets the requirements in DeVnition 10.2. For any ϕ ∈ Dh by the Lebesgue dominated convergence theorem we have 

 ∂h ϕ(x) f (x)μ(dx) = lim

∂h ϕ(x) f n (x)μ(dx),

n→∞

X

X



 ϕ(x) f (x)βh (x)μ(dx) = lim

n→∞

X

ϕ(x) f n (x)βh (x)μ(dx). X

Therefore, the integrals of ϕ with respect to the measures ( U f n , h) H converge. Moreover, the limit is the integral of ϕ with respect to ( U f, h) H , which follows by the one-dimensional case applied to the conditional measures. This completes the proof in the case of S BV and bounded { f n }. In the case of BV it is necessary to show that U f has bounded variation. The measures f n possess H -valued vector densities Rn with respect to some common nonnegative measure ν and the sequence of funcshow that for every Borel tions |Rn | H is bounded in L 1 (ν). It sufVces to k vi h i , where h i ∈ H are mapping v such that |v| H ≤ 1 and v = i=1 constant and orthonormal, we have the estimate k   i=1

vi (x) ( U f, h i ) H (dx) ≤ sup Var( U f n ). X

n

It is readily seen that it is enough to do this for v with functions vi such that vi ∈ Dh i . Indeed, by using convolutions we reduce the general case to the case where the function vi has bounded derivatives of any order along the vector h i . Next, we approximate such functions in L 1 (|( U f, h i ) H |) by their products with functions wi,n ∈ Dh i with the following properties: 0 ≤ wi,n ≤ 1, wi,n (y + th i ) = 0 whenever the length δ y,h,i of U y,h i is less than 4/n and otherwise wi,n = 1 on the inner interval of length δ y,h i − 2/n with the same center as U y,h i .

48 Vladimir I. Bogachev

For such functions we have k  k    vi (x) ( U f, h i ) H (dx) = lim vi (x) ( U f n , h i ) H (dx) i=1

n→∞

X

X

i=1 (v(x), Rn (x)) H ν(dx) = lim n→∞ X  ≤ sup |Rn (x)| H ν(dx) n

X

= sup Var U f n . n

Thus, the case of a uniformly bounded { f n } is considered. Let us now proceed to the general case where { f n } is not uniformly bounded. Take a smooth increasing function ψ on the real line such that ψ(t) = t if |t| ≤ 1, ψ(t) = 2 sign t if |t| ≥ 3, |ψ(t)| ≤ |t|, and |ψ  (t)| ≤ 1. Let ψm (t) = ψ(t/m). According to Lemma 10.4, for every Vxed m the functions ψm ( f n ) belong to the respective (SBV or BV) class and their norms are uniformly bounded in n and m. Hence the function ψm ( f ) belongs to the same class and its norm does not exceed the supremum of the norms of { f n }. We shall deal with a Borel version of f , so the functions ψm ( f ) are also Borel. The Borel sets Bm = {| f | < m} are increasing to X and |ψm ( f )| ≤ | f |. Clearly, f ∈ M H (U, μ) and  f  M = lim ψm ( f ) M . Since ψm+1 ( f ) coincides with ψm ( f ) on m→∞

the set Bm , we obtain that the conditional measures ( U ψm ( f ), h)x,h H on the straight lines x + Rh have a Vnite setwise limit for almost every x, and the measures σ x,h obtained in the limit give rise to bounded measures σ x,h μ(dx), which can be taken for ( U f, h) H . In the case of BVH (U, μ) we have additionally that the resulting vector measure f is of bounded variation. Corollary 10.7. If f ∈ S BVH (U, μ) and ψ is a Lipschitzian function on the real line, then ψ( f ) ∈ S BVH (U, μ) and ψ( f ) S BV ≤ C f  S BV , where C is a Lipschitz constant for ψ. The same is true in the case of BVH (U, μ). Proof. For smooth ψ this has already been noted. The general case follows by approximation and the above theorem. Theorem 10.8. Suppose that IU ∈ S BVH (μ). Then for every function f ∈ S BVH (U, μ) ∩ L ∞ (U, μ)

49 Sobolev classes on infinite-dimensional spaces

its extension by zero outside of U gives a function in the class S BVH (μ). In the opposite direction, the restriction to U of every function in S BVH (μ), not necessarily bounded, gives a function in S BVH (U, μ). If IU ∈ BVH (μ), then the analogous assertions are true for the class BVH (U, μ). Proof. Let f ∈ S BVH (U, μ) ∩ L ∞ (U, μ). We may assume that | f | ≤ 1. Let us Vx h ∈ H . Then we can Vnd a version of f whose restrictions to the straight lines x +Rh have locally bounded variation. Let ax be an endpoint of the interval Ux,h (if it exists). Then the considered version of f has a limit at ax (left or right, respectively), bounded by 1 in the absolute value. DeVned by zero outside of U , the function f remains a function of locally bounded variation on all these straight lines, but at the endpoints of Ux,h its generalized derivative may gain Dirac measures with coefVcients bounded by 1 in the absolute value. However, such Dirac measures (with the coefVcient 1 at the left end and the coefVcient −1 at the right end) are already present in the derivative of the restriction of , we obtain IU . Thus, after adding these point measures to ( U f, h)x,h,μ H a measure that differs from ( U f, h) H by some measure with semivariation not exceeding ( IU , h) H , hence is also of bounded semivariation. Therefore, Lemma 8.6 gives the inclusion of the extension to S BVH (μ). The fact that f |U ∈ S BVH (U, μ) for any f ∈ S BVH (μ) follows by Lemma 10.3, since the restriction of f to U serves as U f . In the case of BVH (μ) we also use the fact that any H -valued measure of bounded variation is given by a Bochner integrable vector density with respect to a suitable scalar measure. Proposition 10.9. Let U be a Borel convex set. Then IU ∈ S BVH (μ). Proof. Let us Vx h ∈ H . If Ux,h is not empty and not the whole straight line, the generalized derivative σ x,h of the function t → IU (x + th) is either the difference of two Dirac’s measures at the endpoints of Ux,h or Dirac’s measure (with the sign plus or minus) at the single endpoint (if Ux,h is a ray). Let us deVne ( IU , h)x,h H by x,h x,h  . ( IU , h)x,h H := σ x,h  ≤ 2, We obtain a bounded measure ( IU , h)x,h H μ(dx) (indeed, σ x,h x,h 0 ≤ IUx,h ≤ 1,   and ∂t   are μ-integrable), which deVnes an H -valued measure IU with the properties mentioned in Lemma 8.6, which yields the desired conclusion.

It should be noted that even for a bounded convex Borel set U in a separable Hilbert space the indicator function IU does not always belong

50 Vladimir I. Bogachev

to BVH (γ ); explicit examples are given in Caselles, Lunardi, Miranda, Novaga [27]. On the other hand, the indicator of any open convex set in the Gaussian case belongs to BVH (γ ) (see [27]). It is worth noting that it is also of interest to consider restrictions of Sobolev functions to Vnite-dimensional subspaces and to inVnite-dimensional surfaces, see Airault, Bogachev, Lescot [3], Celada, Lunardi [29]. Finally, note that a number of other interesting topics related to Sobolev and BV classes on inVnite-dimensional spaces have not been mentioned in this survey. In particular, Cruzeiro [32] initiated the study of the continuity equation and related transformations of measures in inVnite dimensions, and this direction is further developed by many authors, see Ambrosio, Figalli [5], Bogachev, Mayer-Wolf [19], Kolesnikov, R¨ockner ¨ unel, Zakai [87]; Sobolev classes can be useful in [60], Peters [75], Ust¨ the study of the Monge–Kantorovich problem in inVnite dimensions, see ¨ unel [44] and the subsequent papers Bogachev, Kolesnikov Feyel, Ust¨ [18], Cavalletti [28], Fang, Shao, Sturm [42]. Differential equations for Sobolev functions on Hilbert spaces are considered in Da Prato [33]. Additional references on diverse aspects of Sobolev classes can be found in Bogachev [16] and [17].

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[7] L. A MBROSIO and S. D I M ARINO, Equivalent deFnitions of BV space and of total variation on metric measure spaces. J. Funct. Anal. 2014. 266 (2014), 4150–4188. [8] L. A MBROSIO , M. M IRANDA , S. M ANIGLIA and D. PALLARA, Towards a theory of BV functions in abstract Wiener spaces. Physica D: Nonlin. Phenom. 2010. 239 (2010), n. 15, 1458–1469. [9] L. A MBROSIO , M. M IRANDA (J R .), S. M ANIGLIA and D. PAL LARA, BV functions in abstract Wiener spaces, J. Funct. Anal. 2010. 258 (2010), n. 3, 785–813. [10] L. A MBROSIO and P. T ILLI, “Topics on Analysis in Metric Spaces”, Oxford University Press, Oxford, 2004. [11] S. B LANCHERE , D. C HAFAI , F. F OUGERES , I. G ENTIL , F. M ALRIEN , C. ROBERTO and G. S CHEFFER, “Sur les in´egalit´es de Sobolev logarithmiques. Panoramas et Synth`eses”, Soc. Math. France, 2000. [12] V. I. B OGACHEV, Differentiable measures and the Malliavin calculus. J. Math. Sci. (New York) 87 (1997), n. 4, 3577–3731. [13] V. I. B OGACHEV, “Gaussian Measures”, Amer. Math. Soc., Providence, Rhode Island, 1998. [14] V. I. B OGACHEV, Extensions of H -Lipschitzian mappings with inFnite-dimensional range, InVn. Dim. Anal., Quantum Probab. Relat. Top. 2 (1999), n. 3, 1–14. [15] V. I. B OGACHEV, “Measure Theory”, Vol. 1, 2, Springer, Berlin, 2007. [16] V. I. B OGACHEV, “Differentiable Measures and the Malliavin Calculus”, Amer. Math. Soc., Providence, Rhode Island, 2010. [17] V. I. B OGACHEV, Gaussian measures on inFnite-dimensional spaces, In: “Real and Stochastic Analysis, Current Trends”, M. M. Rao (ed.), World Sci., Singapore, 2014, 1–83. [18] V. I. B OGACHEV and A. V. KOLESNIKOV, The Monge-Kantorovich problem: achievements, connections, and perspectives, Russian Math. Surveys. 67 (2012), n. 5, 3–110. [19] V. I. B OGACHEV and E. M AYER -W OLF, Absolutely continuous Gows generated by Sobolev class vector Felds in Fnite and inFnite dimensions, J. Funct. Anal. 167 (1999), n. 1, 1–68. [20] V. I. B OGACHEV, A. Y U . P ILIPENKO and E. A. R EBROVA, Classes of funtions of bounded variation on inFnite-dimensional domains. Dokl. Russian Acad. Sci. 451 (2013), n. 2., 127–131 (in Russian); English transl.: Dokl. Math. 88 (2013), n. 1, 391–395. [21] V. I. B OGACHEV, A. Y U . P ILIPENKO and A. V. S HAPOSHNIKOV, Sobolev functions on inFnite-dimensional domains, J. Math. Anal. Appl. 419 (2014), 1023–1044.

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ASZ and P. KOSKELA, “Sobolev met Poincar´e”, Mem. Amer. Math. Soc., 2000, Vol. 145, n. 688. [54] J. H EINONEN, “Lectures on Analysis on Metric Spaces”, Springer, New York, 2001. [55] M. H INO, Integral representation of linear functionals on vector lattices and its application to BV functions on Wiener space, In: “Stochastic Analysis and Related Topics in Kyoto”, Adv. Stud. Pure Math., Vol. 41, Math. Soc. Japan, Tokyo, 2004, 121–140. [56] M. H INO, Sets of Fnite perimeter and the Hausdorff–Gauss measure on the Wiener space. J. Funct. Anal. 258 (1010), n. 5, 1656– 1681. [57] M. H INO, Dirichlet spaces on H -convex sets in Wiener space, Bull. Sci. Math. 135 (2011), 667–683. Erratum: ibid. 137 (2013), 688– 689. [58] S. JANSON, “Gaussian Hilbert Spaces”, Cambridge Univ. Press, Cambridge, 1997. [59] S. K EITH, A differentiable structure for metric measure spaces, Adv. Math. 183 (2004), n. 2, 271–315. ¨ , On continuity equations in [60] A. KOLESNIKOV and M. R OCKNER inFnite dimensions with non-Gaussian reference measure, J. Funct. Anal. 266 (2014), n. 7, 4490–4537. [61] M. K R E´ E, Propri´et´e de trace en dimension inFnie, d’espaces du type Sobolev, C. R. Acad. Sci., S´er. A 297 (1974), 157–164. [62] M. K R E´ E, Propri´et´e de trace en dimension inFnie, d’espaces du type Sobolev, Bull. Soc. Math. France. 105 (1977), n. 2, 141–163. [63] B. L ASCAR, Propri´et´es locales d’espaces de type Sobolev en dimension inFnie. Comm. Partial Differential Equations 1 (1976), n. 6, 561–584. [64] E. H. L IEB and M. L OSS, “Analysis”, Amer. Math. Soc., Providence, Rhode Island, 2001. [65] P. M ALLIAVIN, Stochastic calculus of variation and hypoelliptic operators, Proc. Intern. Symp. on Stoch. Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976), Wiley, New York – Chichester – Brisbane, 1978, 195–263. [66] P. M ALLIAVIN, “Stochastic Analysis”, Springer-Verlag, Berlin, 1997. [67] P. M ALLIAVIN and A. T HALMAIER, “Stochastic Calculus of Variations in Mathematical Finance”, Springer-Verlag, Berlin, 2006. [68] J. NASH, Continuity of solutions of parabolic and elliptic equations. Amer. J. Math. 80 (1958), 931–954.

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orem, Trends in probability and related analysis (Taipei, 1996), World Sci. Publ., River Edge, New Jersey, 1997, 273–288. I. S HIGEKAWA, “Stochastic Analysis”, Amer. Math. Soc., Providence, Rhode Island, 2004. ¨ and F. S ORIA, Sharp estimates for the non-centered P. S J OGREN maximal operator associated to Gaussian and other radial measures, Adv. Math. 181 (2004), n. 2, 251–275. A. V. S KOROHOD, “Integration in Hilbert Space”, Springer-Verlag, Berlin – New York, 1974. A. J. S TAM, Some inequalities satisFed by the quantities of information of Fisher and Shannon, Information and Control 2 (1959), 101–112. ¨ ST UNEL ¨ and M. Z AKAI, “Transformation of Measure on A. S. U Wiener Space”, Springer-Verlag, Berlin, 2000. S. K. VODOP ’ YANOV, Topological and geometric properties of mappings with an integrable Jacobian in Sobolev classes. I, Sibirsk. Mat. Zh. 41 (2000), n. 1, 23–48 (in Russian); English transl.: Siberian Math. J. 41 (2000), n. 1, 19–39. S. WATANABE, Fractional order Sobolev spaces on Wiener space, Probab. Theory Relat. Fields 95 (1993), 175–198. V. V. Z HIKOV, Weighted Sobolev spaces, Matem. Sbornik 189 (1998), n. 8, 27–58 (in Russian); English transl.: Sbornik Math. 189 (1998), n. 8, 1139–1170. V. V. Z HIKOV, On the density of smooth functions in a weighted Sobolev space, Dokl. Russian Acad. Sci. 453 (2013), n. 3, 247–251 (in Russian); English transl.: Dokl. Math. 88 (2013), n. 3, 669–673. W. Z IEMER, “Weakly Differentiable Functions”, Springer-Verlag, New York – Berlin, 1989.

Isoperimetric problem and minimal surfaces in the Heisenberg group Roberto Monti

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ?? 1. Introduction to the Heisenberg group Hn . . . . . . . . . . . . . . . . . . . . . . 58 1.1. Algebraic structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 1.2. Metric structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 2. Heisenberg perimeter and other equivalent measures . . . . . . . . . . . 66 2.1. H -perimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.2. Equivalent notions for H -perimeter . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.3. RectiVability of the reduced boundary . . . . . . . . . . . . . . . . . . . . . . 76 3. Area formulas, Vrst variation and H -minimal surfaces . . . . . . . . . 78 3.1. Area formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.2. First variation and H -minimal surfaces . . . . . . . . . . . . . . . . . . . . . 85 3.3. First variation along a contact Wow . . . . . . . . . . . . . . . . . . . . . . . . . 94 4. Isoperimetric problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.1. Existence of isoperimetric sets and Pansu’s conjecture . . . . . . . . 99 4.2. Isoperimetric sets of class C 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.3. Convex isoperimetric sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.4. Axially symmetric solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.5. Calibration argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 5. Regularity problem for H -perimeter minimizing sets . . . . . . . . . 113 5.1. Existence and density estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.2. Examples of nonsmooth H -minimal surfaces . . . . . . . . . . . . . . . 117 5.3. Lipschitz approximation and height estimate . . . . . . . . . . . . . . . 123

This text is an extended version of the lecture notes of the course Isoperimetric problem and minimal surfaces in the Heisenberg group given at the ERC-School on Geometric Measure Theory and Real Analysis, held in Pisa between 30th September and 4th October 2013.

58 Roberto Monti

1 Introduction to the Heisenberg group Hn 1.1 Algebraic structure The 2n + 1-dimensional Heisenberg group is the manifold Hn = Cn × R, n ∈ N, endowed with the group product   (z, t) · (ζ, τ ) = z + ζ, t + τ + 2 Imz, ζ¯  , (1.1) where t, τ ∈ R, z, ζ ∈ Cn and z, ζ¯  = z 1 ζ¯1 + . . . + z n ζ¯n . The Heisenberg group is a noncommutative Lie group. The identity element is 0 = (0, 0) ∈ Hn . The inverse element of (z, t) is (−z, −t). The center of the group is Z = {(z, t) ∈ Hn : z = 0}. We denote elements of Hn by p = (z, t) ∈ Cn × R. The left translation by p ∈ Hn is the mapping L p : Hn → Hn L p (q) = p · q. Left translations are linear mappings in Hn = R2n+1 . For any λ > 0, the mapping δλ : Hn → Hn δλ (z, t) = (λz, λ2 t),

(1.2)

is called dilation. Dilations are linear mappings and form a 1-parameter group (δλ )λ>0 of automorphisms of Hn . We denote by |E| the Lebesgue measure of a Lebesgue measurable set E ⊂ Hn = R2n+1 . The differential d L p of any left translation is an upper triangular matrix with 1 along the principal diagonal. It follows that det d L p = 1 on Hn for any p ∈ Hn and, as a consequence, |L p E| = |E|,

for any p ∈ Hn and for any E ⊂ Hn .

Lebesgue measure is the Haar measure of the Heisenberg group. Moreover, we have det δλ = λ Q , where the integer Q = 2n + 2

(1.3)

is called homogeneous dimension of Hn . As a consequence, we have |δλ E| = λ Q |E|. We introduce the Lie algebra of left invariant vector Velds of Hn . A C ∞ vector Veld X in Hn is left invariant if for any function f ∈ C ∞ (Hn ) and for any p ∈ Hn there holds X ( f ◦ L p ) = (X f ) ◦ L p .

59 Isoperimetric problem and minimal surfaces

Equivalently, X is left invariant if X ( p) = d L p X (0), where d L p is the differential of the left translation by p. Left invariant vector Velds with the bracket form a nilpotent Lie algebra hn , called Heisenberg Lie algebra. The algebra hn is spanned by the vector Velds Xj =

∂ ∂ + 2y j , ∂x j ∂t

Yj =

∂ ∂ − 2x j , ∂yj ∂t

and

T =

∂ , (1.4) ∂t

with j = 1, . . . , n. In other words, any left invariant vector Veld is a linear combination with real coefVcients of the vector Velds (1.4). We are using the notation p = (z, t) and z = x + iy with x, y, ∈ Rn . The vector Velds (1.4) are determined by the relations ∂ , ∂x j ∂ , Y j ( p) = d L p Y j (0) = d L p ∂yj ∂ T ( p) = d L p T (0) = d L p . ∂t

X j ( p) = d L p X j (0) = d L p

The distribution of 2n-dimensional planes H p spanned by the vector Velds X j and Y j , j = 1, . . . , n, is called horizontal distribution:   (1.5) H p = span X j ( p), Y j ( p) : j = 1, . . . , n . The horizontal distribution is nonintegrable. In fact, for any j = 1, . . . , n there holds (1.6) [X j , Y j ] = −4T = 0. All other commutators vanish. The horizontal distribution is bracket generating of step 2. When n = 1, we write X = X 1 and Y = Y1 . 1.2 Metric structure We introduce the Carnot-Carath´eodory metric of Hn and we describe the geodesics of this metric. In H1 , these curves are important in the structure of H -minimal surfaces and surfaces with constant H -curvature. A Lipschitz curve γ : [0, 1] → Hn is horizontal if γ˙ (t) ∈ Hγ (t) for a.e. t ∈ [0, 1]. Equivalently, γ is horizontal if there exist functions h j ∈ L ∞ ([0, 1]), j = 1, . . . , 2n, such that γ˙ =

n  j=1

h j X j (γ ) + h n+ j Y j (γ ),

a.e. on [0, 1].

(1.7)

60 Roberto Monti

The coefVcients h j are unique, and by the structure of the vector Velds X j and Y j they satisfy h j = γ˙ j , where γ = (γ1 , . . . , γ2n+1 ) are the coordinates of γ given by the identiVcation Hn = R2n+1 . We call the Lipschitz curve κ : [0, 1] → R2n , κ = (γ1 , . . . , γ2n ), horizontal projection of γ . The vertical component of γ is determined by the horizontality condition (1.7). Namely, we have γ˙2n+1 = 2

n 

h j γn+ j − h n+ j γ j = 2

j=1

n 

κ˙ j κn+ j − κ˙ n+ j κ j ,

j=1

and, by integrating, we obtain for any t ∈ [0, 1] n  t  γ2n+1 (t) = γ2n+1 (0) + 2 (κ˙ j κn+ j − κ˙ n+ j κ j )ds. j=1

(1.8)

0

If κ is a given Lipschitz curve in R2n , the curve γ with (γ1 , . . . , γ2n ) = κ and γ2n+1 as in (1.8) is called a horizontal lift of κ and we write γ = Lift(κ). The horizontal lift is unique modulo the initial value γ2n+1 (0). Now we deVne the Carnot-Carath´eodory metric of Hn . For any pair of points p, q ∈ Hn , there exists a horizontal curve γ : [0, 1] → Hn such that γ (0) = p and γ (1) = q. This follows from the nonintegrability condition (1.6) and it can be checked via a direct computation. The basic observation is that for any t ∈ R exp(−tY j ) exp(−t X j ) exp(tY j ) exp(t X j )(0, 0) = (0, −4t 2 ), where exp(t V )( p) is the Wow of the vector Veld V at time t starting from p. We Vx on the horizontal distribution H p the positive quadratic form g( p; ·) making the vector Velds X 1 , . . . , X n , Y1 , . . . , X n orthonormal at every point p ∈ Hn . Since the vector Velds are left invariant, the quadratic form is left invariant. We use the quadratic form g to deVne the length of a horizontal curve γ : [0, 1] → Hn with horizontal projection κ:  1  1 g(γ ; γ˙ )1/2 dt = |κ|dt, ˙ L(γ ) = 0

0

˙ For any couple of points where |κ| ˙ is the Euclidean norm in R2n of κ. p, q ∈ Hn , we deVne d( p, q)  (1.9)

= inf L(γ ) : γ : [0, 1] → Hn is horizontal, γ (0) = p and γ (1) = q .

61 Isoperimetric problem and minimal surfaces

We already observed that the above set is nonempty for any p, q ∈ Hn , and thus 0 ≤ d( p, q) < ∞. The function d : Hn × Hn → [0, ∞) is a distance on Hn , called Carnot-Carath´eodory distance. It can be proved that for any compact set K ⊂ Hn = R2n+1 there exists a constant 0 < C K < ∞ such that d( p, q) ≥ C K | p − q|

(1.10)

for all p, q ∈ K , where | p − q| is the Euclidean distance between the points. In particular, we have d( p, q) = 0 if p  = q. The distance d is left invariant and 1-homogeneous. Namely, for any p, q, w ∈ Hn and λ > 0 there holds: i) d(w · p, w · q) = d( p, q); ii) d(δλ ( p), δλ (q)) = d( p, q). Statement i) follows from the fact that L(w · γ ) = L(γ ) for any horizontal curve γ and for any w ∈ Hn , because the quadratic form g is left invariant. Analogously, ii) follows from L(δλ (γ )) = λL(γ ), that is a consequence of the identities X j ( f ◦ δλ ) = λ(X j f ) ◦ δλ ,

Y j ( f ◦ δλ ) = λ(Y j f ) ◦ δλ ,

holding for any f ∈ C ∞ (Hn ) and λ > 0. For any p ∈ Hn and r > 0, we deVne the Carnot-Carath´eodory ball   Br ( p) = q ∈ Hn : d( p, q) < r . We also let Br = Br (0). The size of Carnot-Carath´eodory balls can be described by means of anisotropic homogeneous norms. For any p = (z, t) ∈ Hn let (1.11)  p∞ = max{|z|, |t|1/2 }. The “box norm”  · ∞ has the following properties: i) δλ ( p)∞ = λ p∞ , for all p ∈ Hn and λ > 0; ii)  p · q∞ ≤  p∞ + q∞ , for all p, q ∈ Hn . By ii), the function  : Hn × Hn → [0, ∞), ( p, q) =  p−1 · q∞ ,

(1.12)

satisVes the triangle inequality and is a distance on Hn . By an elementary argument based on continuity, compactness, and homogeneity, there exists an absolute constant C > 0 such that C −1 d( p, q) ≤ ( p, q) ≤ Cd( p, q) for all p, q ∈ Hn . The distance functions d and  are equivalent.

62 Roberto Monti

All the previous observations are still valid when the “box norm” ·∞  1/4 is replaced with the Koranyi norm  p = |z|4 + t 2 . n The metric space (H , d) is complete and locally compact. By the deVnition of d, it is also a length space. Then, a standard application of Ascoli-Arzel`a theorem shows that it is a geodesic space, namely for all p, q ∈ Hn there exists a horizontal curve γ : [0, 1] → Hn such that γ (0) = q, γ (1) = p, and L(γ ) = d( p, q). The curve γ is called geodesic or length minimizing curve joining q to p. We classify geodesics in H1 starting from the initial point 0. Let  : [0, 2π] × [−2π, 2π] → H1 be the mapping (ψ, ϕ) =

eiψ (eiϕ − 1) ϕ

,2

ϕ − sin ϕ

. ϕ2

(1.13)

When ϕ = 0, the formula is determined by analytic continuation and we have (ψ, 0) = (ieiψ , 0). The set S = ([0, 2π] × [−2π, 2π]) ⊂ H1 is homeomorphic to a 2-dimensional sphere. It is a C ∞ surface at points (z, t) ∈ S such that z = 0, i.e., (z, t) = (ψ, ϕ) with |ϕ|  = 2π. The antipodal points (0, ±1/π) ∈ S are obtained for ϕ = ±2π and are Lipschitz points. We will show that S is the unitary Carnot-Carath´eodory sphere of H1 centered at 0, S = ∂ B1 (0). Theorem 1.1. For any ψ ∈ [0, 2π] and ϕ ∈ [−2π, 2π], the curve γψ,ϕ : [0, 1] → H1 γψ,ϕ (s) =

eiψ (eiϕs − 1) ϕs − sin ϕs

, ,2 ϕ ϕ2

s ∈ [0, 1],

(1.14)

is length minimizing. When |ϕ| < 2π, γψ,ϕ is the unique length minimizing curve from 0 to (ψ, ϕ). When ϕ = ±2π, for every ψ ∈ [0, 2π] the curve γψ,ϕ is length minimizing from 0 to (0, ±1/π). Proof. Let (z 0 , t0 ) ∈ H1 be any point and introduce the family of admissible curves

 A = κ ∈ Lip([0, 1]; R2 ) : κ(0) = 0, κ(1) = z 0 . The end-point mapping relative to the third coordinate End : A → R is  End(κ) = 2

1

  κ2 κ˙ 1 − κ1 κ˙ 2 ds = 2

0

where κ κ˙¯ is a complex product.

 0

1

˙¯ Im(κ κ)ds,

63 Isoperimetric problem and minimal surfaces

The geodesic γ joining 0 to (z 0 , t0 ) is the horizontal lift of the curve κ in the plane that solves the problem min



1

 |κ|ds ˙ : κ ∈ A and End(κ) = t0 .

(1.15)

0

Let κ be a minimizer for problem (1.15). We compute the Vrst variation of the length functional at the curve κ with constraint End(κ) = t0 . For  τ ∈ R and ϑ ∈ Cc∞ (0, 1); R2 the curve κ τ = κ + τ ϑ satisVes   1   d d τ  (κ2 + τ ϑ2 )(κ˙ 1 + τ ϑ˙ 1 ) =2 End(κ ) dτ dτ 0 τ =0    − (κ1 + τ ϑ1 )(κ˙ 2 + τ ϑ2 ) ds  

=2

τ =0

1

  ϑ2 κ˙ 1 + κ2 ϑ˙ 1 − κ1 ϑ2 − ϑ1 κ˙ 2 ds

1

  ϑ2 κ˙ 1 − ϑ1 κ˙ 2 ds.

0

 =4

0

  We have |κ| ˙  = 0 a.e., and thus there exists ϑ ∈ Cc∞ (0, 1); R2 such that   d τ  = 0. (1.16) End(κ ) dτ τ =0   Fix a function ϑ satisfying (1.16) and let η ∈ Cc∞ (0, 1); R2 be an arbitrary vector valued function. The curve κ +τ ϑ +η belongs to A. DeVne the function in the plane E : R2 → R E(, τ ) = End(κ + τ ϑ + η). This function is C 1 -smooth and H := ∂ E(0, 0)/∂τ  = 0, by (1.16). By the implicit function theorem, there exist 0 > 0 and a function τ ∈ C 1 (−0 , 0 ) such that E(, τ ()) = E(0, 0) = t0 for all  ∈ (−0 , 0 ). Moreover, we have ∂ E(0, 0) −1 ∂ E(0, 0)

τ  (0) = − ∂τ ∂  1  1 1 ⊥ 1 (η2 κ˙ 1 − η1 κ˙ 2 )ds = − κ˙ , ηds, =− H 0 H 0

(1.17)

˙ where κ˙ ⊥ = (−κ˙ 2 , κ˙ 1 ), or equivalently, in the complex notation κ˙ ⊥ = i κ.

64 Roberto Monti

Since κ is a solution to the minimum problem (1.15) and κ + τ ()ϑ + η ∈ A with End(κ + τ ()ϑ + η) = t0 , then we have 

1

1=



1

|κ|ds ˙ ≤

0

|κ˙ + τ ()ϑ˙ +  η|ds ˙ = L(),

0

and thus L  (0) = 0. We can without loss of generality assume that κ is parameterized by arc-length, i.e., |κ| ˙ = 1. The equation L  (0) = 0 gives (we also use (1.17)) 

 1 ˙ 0 = τ (0) κ, ˙ ϑds + κ, ˙ ηds ˙ 0 0  1  1 =ϕ κ˙ ⊥ , ηds + κ, ˙ ηds, ˙ 

1

0

0

where ϕ ∈ R is the constant 1 ϕ=− H



1

˙ κ, ˙ ϑds.

0

  Eventually, for any test function η ∈ Cc∞ (0, 1); R2 we have 

1

  κ, ˙ η ˙ + ϕκ˙ ⊥ , η ds = 0,

0

and a standard argument implies that κ is in C ∞ ([0, 1]; R2 ) and it solves ˙ Then we have κ(s) ˙ = ieiψ eiϕs , the differential equation κ¨ = ϕ κ˙ ⊥ = iϕ κ. s ∈ R, for some ψ ∈ [0, 2π]. Integrating with κ(0) = 0, we Vnd κ(s) =

eiψ (eiϕs − 1) , ϕ

s ∈ R.

The vertical coordinate of the horizontal lift γ of κ is  s ϕs − sin ϕs Im(κ(σ )κ(σ ˙ ))dσ = 2 , γ3 (s) = 2 ϕ2 0 and thus for any ψ ∈ [0, π] and ϕ ∈ R we get the curve γψ,ϕ (s) =

eiψ (eiϕs − 1) ϕs − sin ϕs

, ,2 ϕ ϕ2

s ∈ R.

When ϕ = 0, γ reduces to the line γ (s) = (ieiψ s, 0).

(1.18)

65 Isoperimetric problem and minimal surfaces

The curve γψ,ϕ is length minimizing on the interval 0 ≤ s ≤ 2π/|ϕ| and, after s = 2π/|ϕ|, it ceases to be length minimizing. We prove this claim in the case ϕ = 2π by a geometric argument. For s = 1 we have   γψ,2π (1) = 0, 1/π . At the point (0, 1/π) ∈ C × R, the surface S = ([0, 2π] × [−2π, 2π]) introduced in (1.13) has a conical point directed downwards. By this, we mean that near (0, 1/π) the surfaces S stays above the cone t = 1/π + δ|z| for some δ > 0. Then for any  > 0 small enough there exist ¯ ϕ) 0 < λ < 1 and (ψ, ¯ ∈ [0, 2π] × [−2π, 2π] such that γψ,2π (1 + ) = ¯ ϕ). ¯ ϕ), ¯ Since d((ψ, ¯ 0) ≤ 1 (a posteriori we have equality, δλ (ψ, here), we deduce that ¯ ϕ), ¯ 0) ≤ λ < 1. d(γψ,2π (1 + ), 0) = λd((ψ, Since the length of γψ,2π on the interval [0, 1 + ] is 1 + , we see that the curve is not length minimizing. For any point (z 0 , t0 ) ∈ H1 with z 0 = 0, the system of equations eiψ (eiϕs − 1) = z0, ϕ

2

ϕs − sin ϕs = t0 , ϕ2

(1.19)

has unique solutions s ≥ 0, ψ ∈ [0, 2π), and ϕ ∈ R subject to the constraint s|ϕ| < 2π (we omit details). Thus γψ,ϕ is the unique length minimizing curve from 0 to (z 0 , t0 ) and s = d((z 0 , t0 ), 0). Remark 1.2. The Heisenberg isoperimetric problem is related to the classical Dido problem, that asks to bound a region of the half plane with a curve with minimal length, where the boundary of the half plane (the coast) is a free length. Let γ : [0, 1] → H1 be a horizontal curve such that γ (0) = 0 and let κ : [0, 1] → R2 be its horizontal projection. By formula (1.8), the third coordinate of γ at time t ∈ [0, 1] is   t  ydx − xdy. γ3 (t) = 2 (κ2 κ˙ 1 − κ1 κ˙ 2 ds = 2 κ|[0,t]

0

Let E t ⊂ R2 be the region of the plane bounded by the curve κ restricted to [0, t] and by the line segment joining κ(t) to 0. Assume that the concatenation of κ and of the line segment bounds E t counterclockwise. Then by Stokes’ theorem we have  dx ∧ dy = −4|E t |. γ3 (t) = −4 Et

66 Roberto Monti

If the orientation is clockwise, −4|E t | is replaced by 4|E t |. If the orientation is different in subregions of E t , there are area cancellations. So the minimum problem (1.15) consists in Vnding the shortest curve in the plane enclosing an amount of area given by the t0 coordinate of the Vnal point (z 0 , t0 ). In the Heisenberg isoperimetric problem, the point z 0 is also Vxed, differently from Dido problem.

2 Heisenberg perimeter and other equivalent measures 2.1 H -perimeter We introduce the notion of H -perimeter for a set E ⊂ Hn . We preliminarily need the deVnition of H -divergence of a vector valued function ϕ ∈ C 1 (Hn ; R2n ). Let V be a smooth vector Veld in Hn = R2n+1 . We may express V using both the basis X j , Y j , T and the standard basis of vector Velds of R2n+1 : n    V = ϕ j X j + ϕn+ j Y j + ϕ2n+1 T j=1

n   ∂ ∂ ∂ ∂ + ϕn+ j + 2y j ϕ j − 2x j ϕn+ j ϕj + ϕ2n+1 , = ∂ x ∂ y ∂t ∂t j j j=1

(2.1)

where ϕ j , ϕn+ j , ϕ2n+1 ∈ C ∞ (Hn ) are smooth functions. The standard divergence of V is div V =

∂ϕ j ∂ϕn+ j  ∂ϕn+ j + + 2y j − 2x j ∂x j ∂yj ∂t ∂t

n  ∂ϕ j j=1

∂ϕ2n+1 ∂t n    = X j ϕ j + Y j ϕn+ j + T ϕ2n+1 . +

(2.2)

j=1

The vector Veld V is said to be horizontal if V ( p) ∈ H p for all p ∈ Hn . Namely, a vector Veld V as in (2.1) is horizontal when ϕ2n+1 = 0. These observations suggest the following deVnition. Let A ⊂ Hn be an open set. We deVne the horizontal divergence of a vector valued mapping ϕ ∈ C 1 (A; R2n ) as divH ϕ =

n   j=1

 X j ϕ j + Y j ϕn+ j .

(2.3)

67 Isoperimetric problem and minimal surfaces

By (2.2), divH ϕ = div V is the standard divergence of the horizontal vector Veld V with coordinates ϕ = (ϕ1 , . . . , ϕ2n ) in the basis X 1 , . . . , X n , Y1 , . . . , Yn . If  ·  is the norm on H p that makes X 1 , . . . , X n , Y1 , . . . , Yn orthonormal, then we have V ( p) = |ϕ( p)|, where | · | is the standard norm on R2n . The following deVnition is the starting point of the fundamental paper [27] (see also [33]). DeHnition 2.1 (H -perimeter). The H -perimeter in an open set A ⊂ Hn of a Lebesgue measurable set E ⊂ Hn is   P(E; A) = sup divH ϕ dzdt : ϕ ∈ Cc1 (A; R2n ), ϕ∞ ≤ 1 . (2.4) E

Above, we let

ϕ∞ = sup |ϕ( p)|. p∈A

If P(E;A) < ∞, we say that E has Vnite H -perimeter in A. If P(E;A ) < ∞ for any open set A  A, we say that E has locally Vnite H -perimeter in A. H -perimeter has the following invariance properties. Proposition 2.2. Let E ⊂ Hn be a set with Fnite H -perimeter in an open set A ⊂ Hn . Then for any p ∈ Hn and for any λ > 0 we have: i) P(L p E; L p A) = P(E; A); ii) P(δλ E; δλ A) = λ Q−1 P(E; A). Proof. Statement i) follows from the fact that the vector Velds X j and Y j are left invariant, and thus (divH ϕ) ◦ L p = divH (ϕ ◦ L p ). We prove ii) in the case A = Hn . First notice that for any ϕ ∈ Cc1 (Hn ; R2n ) we have divH (ϕ ◦ δλ ) = λ(divH ϕ) ◦ δλ , and thus    Q Q−1 divH ϕ dzdt = λ (divH ϕ) ◦ δλ dzdt = λ divH (ϕ ◦ δλ )dzdt. δλ E

E

The claim easily follows.

E

68 Roberto Monti

Let E ⊂ Hn be a set with locally Vnite H -perimeter in an open set A ⊂ Hn . The linear functional T : Cc1 (A; R2n ) → R  divH ϕ(z, t) dzdt T (ϕ) = E

is locally bounded in Cc (A; R2n ). Namely, for any open set A  A we have T (ϕ) ≤ ϕ∞ P(E; A ) (2.5) for all ϕ ∈ Cc1 (A ; R2n ). By density, T can be extended to a bounded linear operator on Cc (A ; R2n ) satisfying the same bound (2.5). Thus, by Riesz’ representation theorem we deduce the following proposition. Proposition 2.3. Let E ⊂ Hn be a set with locally Fnite H -perimeter in the open set A ⊂ Hn . There exist a positive Radon measure μ E on A and a μ E -measurable function ν E : A → R2n such that: 1) |ν E | = 1 μ E -a.e. on A. 2) The following generalized Gauss-Green formula holds   divH ϕ dzdt = − ϕ, ν E dμ E E

(2.6)

A

for all ϕ ∈ Cc1 (A; R2n ). Above, ·, · is the standard scalar product in R2n . DeHnition 2.4 (Horizontal normal). The measure μ E is called H -perimeter measure and the function ν E is called measure theoretic inner horizontal normal of E. We shall refer to ν E simply as to the horizontal normal. In Section 3, we describe geometrically ν E in the smooth case (see (3.3)). In Proposition 2.10 below, we shall see that the vertical hyperplane in Hn orthogonal to ν E ( p) is the “tangent plane” to ∂ E at points of the reduced boundary. Remark 2.5. By Proposition 2.3, the open sets mapping A → P(E; A ), with A  A open, extends to the Radon measure μ E . We show that for any open set A  A we have μ E (A ) = P(E; A ). The inequality P(E; A ) ≤ μ E (A ) follows from the sup-deVnition (2.4) of H -perimeter. The opposite inequality can be proved by a standard approximation argument. By Lusin’s theorem, for any  > 0 there exists a compact set K ⊂ A such that μ E (A \K ) <  and ν E : K → R2n is continuous. By Titze’s theorem, there exists ψ ∈ Cc (A ; R2n ) such that ψ = ν E on K and ψ∞ ≤ 1. Finally, by molliVcation there exists

69 Isoperimetric problem and minimal surfaces

ϕ ∈ Cc∞ (A ; R2n ) such that ϕ − ψ∞ <  and ϕ∞ ≤ 1. Then we have    divH ϕdzdt = − ϕ, ν E dμ E ≥ (1 − )μ E (A ) − 2, P(E; A ) ≥ A

E

and the claim follows. In the sequel, we need a metric structure on Hn . For most purposes, the Carnot-Carath´eodory metric would be Vne. In some cases, however, as in the characterization (2.13) of H -perimeter by means of spherical Hausdorff measures, the structure of Carnot-Carath´eodory balls is less manageable. For this reason, we closely follow [27] and we use the metric  introduced in (1.12) via the “box-norm”  · ∞ in (1.11). We denote the open ball in  centered at p ∈ Hn and with radius r > 0 in the following way   (2.7) U ( p, r) = q ∈ Hn :  p−1 · q∞ < r . We also let Ur ( p) = U ( p, r) and Ur = Ur (0). DeHnition 2.6 (Measure theoretic boundary). The measure theoretic boundary of a measurable set E ⊂ Hn is the set   ∂ E = p ∈ Hn : |E ∩ Ur ( p)| > 0 and |Ur ( p) \ E| > 0 for all r > 0 . The measure theoretic boundary is a subset of the topological boundary. The deVnition does not depend on the speciVc balls Ur ( p). We may also consider the set of points with density 1/2:

|E ∩ Ur ( p)| 1 E 1/2 = p ∈ Hn : lim = . r→0 |Ur ( p)| 2 We clearly have E 1/2 ⊂ ∂ E. The deVnition of E 1/2 is sensitive to the choice of the metric. The perimeter measure μ E is concentrated in a subset of E 1/2 called reduced boundary. The following deVnition is introduced and studied in [27]. DeHnition 2.7 (Reduced boundary). The reduced boundary of a set E ⊂ Hn with locally Vnite H -perimeter is the set ∂ ∗ E of all points p ∈ Hn such that the following three conditions hold: (1) μ E (Ur ( p)) > 0 for all r > 0. (2) We have  lim

r→0

(3) There holds |ν E ( p)| = 1.

Ur ( p)

ν E dμ E = ν E ( p).

70 Roberto Monti

 As usual , stands for the averaged integral. The deVnition of reduced boundary is sensitive to the metric. It also depends on the representative of ν E . The proof of the Euclidean model of Proposition 2.8 below relies upon Lebesgue-Besicovitch differentiation theorem for Radon measures in Rn . In Hn with metrics equivalent to the Carnot-Carath´eodory distance, however, Besicovitch’s covering theorem fails (see [36] and [65]). This problem is bypassed in [27] using an asymptotic doubling property established, in a general context, in [1]. Proposition 2.8. Let E ⊂ Hn be a set with locally Fnite H -perimeter. Then the perimeter measure μ E is concentrated on ∂ ∗ E. Namely, we have μ E (Hn \ ∂ ∗ E) = 0. Proof. By [1], Theorem 4.3, there exists a constant τ (n) > 0 such that for μ E -a.e. p ∈ Hn there holds τ (n) ≤ lim inf r→0

μ E (Ur ( p)) μ E (Ur ( p)) < ∞. ≤ lim sup r Q−1 r Q−1 r→0

As a consequence, we have the following asymptotic doubling formula lim sup r→0

μ E (U2r ( p)) < ∞, μ E (Ur ( p))

(2.8)

for μ E -a.e. p ∈ Hn . Thus, by Theorems 2.8.17 and 2.9.8 in [25], for any function f ∈ L 1loc (Hn ; μ E ) there holds  lim f dμ E = f ( p) r→0

Ur ( p)

for μ E -a.e. p ∈ Hn . Assume that p ∈ Hn \ ∂ ∗ E. There are three possibilities: 1) We have μ E (Ur ( p)) = 0 for some r > 0. The set of points with this property has null μ E measure. 2) We have  lim

r→0

Ur ( p)

ν E dμ E = ν E ( p).

By the above argument with f = ν E , the set of such points has null μ E measure. 3) We have |ν E ( p)| = 1. By Proposition 2.3, the set of such points has null μ E measure. This ends the proof.

71 Isoperimetric problem and minimal surfaces

DeHnition 2.9 (Vertical plane). For any ν ∈ R2n with |ν| = 1, we call the set   Hν = (z, t) ∈ Hn : ν, z ≥ 0, t ∈ R the vertical half-space through 0 ∈ Hn with inner normal ν. The boundary of Hν , the set   ∂ Hν = (z, t) ∈ Hn : ν, z = 0, t ∈ R , is called vertical plane orthogonal to ν passing through 0 ∈ Hn . At points p ∈ ∂ ∗ E, the set E blows up to the vertical half space Hν with ν = ν E ( p). In this sense, the boundary of Hν is the anisotropic tangent space of ∂ ∗ E at p. The problem of the characterization of blowups in Carnot groups is still open. In general, it is known that in the blow-up of blow-ups there are vertical hyperplanes (see [3]). Hereafter, we let E λ = δλ E for λ > 0. Theorem 2.10 (Blow-up). Let E ⊂ Hn be a set with Fnite H -perimeter, assume that 0 ∈ ∂ ∗ E and let ν = ν E (0). Then we have lim χ Eλ = χ Hν ,

(2.9)

λ→∞

where the limit is in L 1loc (Hn ). Moreover, for a.e. r > 0 we have lim P(E λ ; Ur ) = P(Hν ; Ur ) = cn r Q−1 ,

λ→∞

(2.10)

where cn = P(Hν ; U1 ) > 0 is an absolute constant. Proof. Let ϕ ∈ Cc1 (Hn ; R2n ) be a test vector valued function. For a.e. r > 0, we have the following integration by parts formula    divH ϕ dzdt = − ϕ, ν E dμ E − ϕ, νUr dμUr . (2.11) E∩Ur

Ur

∂Ur ∩E

This formula can be proved in the following way. Let ( f j ) j∈N be a sequence of functions f j ∈ C ∞ (Hn ) such that f j → χ E , as j → ∞, in L 1loc (Hn ) and ∇H f j dzdt  ν E dμ E in the weak sense of Radon measures. We are denoting by ∇H f = (X 1 f, . . . , X n f, Y1 f, . . . , Yn f ) the horizontal gradient of a function f . The set Ur supports the standard divergence theorem and therefore we have    f j divH ϕ dzdt = − ϕ, ∇H f j dzdt − f j ϕ, νUr dμUr . (2.12) Ur

Ur

∂Ur

72 Roberto Monti

We can assume that, for a.e. r > 0, f j → χ E in L 1 (∂Ur ) and μ E (∂Ur ) = 0. Letting j → ∞ in (2.12) we obtain (2.11). Let ϕ ∈ Cc1 (Hn ; R2n ) be such that ϕ(z, t) = ν E (0) for all (z, t) ∈ Ur . From (2.11), we have   0=− ν E (0), ν E dμ E − ν E (0), νUr dμUr . ∂Ur ∩E

Ur

Using |ν E (0)| = |νUr | = 1 a.e. and Proposition 2.2, we have   ν E (0), ν E dμ E = − ν E (0), νUr dμUr ≤ P(Ur ; Hn ) ∂Ur ∩E

Ur

= r Q−1 P(U1 ; Hn ). Since 0 ∈ ∂ ∗ E, there holds  ν E (0), ν E dμ E = (1 + o(1))P(E; Ur ), Ur

where o(1) → 0 as r → 0. Using these estimates, we conclude that for any λ ≥ 1 we have P(E λ ; Ur ) = λ Q−1 P(E; Ur/λ ) ≤ 2P(U1 ; Hn )r Q−1 . The family of sets (E λ )λ>1 has locally uniformly bounded perimeter. By the compactness theorem for BVH functions (see [33]), there exists a set F ⊂ Hn with locally Vnite perimeter and a sequence λ j → ∞ such that E λ j → F in the L 1loc (Hn ) convergence of characteristic functions. From the Gauss-Green formula (2.6), it follows that ν Eλ j μ Eλ j  ν F μ F ,

as j → ∞,

in the sense of the weak convergence of Radon measures. Starting from the identity   ν Eλ j dμ Eλ j = ν E dμ E , Ur

Ur/λ j

using 0 ∈ ∂ ∗ E, and choosing r > 0 such that μ F (∂Ur ) = 0 – this holds for a.e. r > 0, – letting j → ∞ we Vnd  ν F , ν E (0)dμ F = 1. Ur

This implies that ν F = ν E (0) μ F -a.e. in Hn , because r > 0 is otherwise arbitrary. By the characterization of sets with constant horizontal normal

73 Isoperimetric problem and minimal surfaces

(see Remark 5.7 below), we have F = Hν with ν = ν(0). We are omitting the proof that 0 ∈ ∂ F. The limit F = Hν is thus independent of the sequence (λ j ) j∈N and this observation concludes the proof of (2.9). We prove (2.10). From   ν, ν Eλ dμ Eλ = ν, ν E dμ E = 1 + o(1), as λ → ∞, Ur

Ur/λ

we deduce that

 P(E λ ; Ur ) = (1 + o(1))

Ur

ν, ν Eλ dμ Eλ .

Letting λ → ∞, using the weak convergence ν Eλ dμ Eλ → ν F dμ F and choosing r > 0 with μ F (∂Ur ) = 0, we get the claim. 2.2 Equivalent notions for H -perimeter In this section, we describe some characterizations of H -perimeter related to the metric structure of Hn . 2.2.1 Hausdorff measures The Heisenberg perimeter has a representation in terms of spherical Hausdorff measures. We use the metric  in (1.12). The diameter of a set K ⊂ Hn is

diam K = sup ( p, q). p,q∈K

If Ur is a ball in the distance  with radius r, then we have diam Ur = 2r. Let E ⊂ Hn be a set. For any s ≥ 0 and δ > 0 deVne the premeasures

  Hs,δ (E) = inf (diam K i )s : E ⊂ K i , K i ⊂ Hn , diam K i < δ , i∈N

i∈N

i∈N

i∈N

 (diam Ui )s : E ⊂

Ss,δ (E) = inf

 Ui ,Ui -balls in Hn , diam Ui < δ ,

Letting δ → 0, we deVne Hs (E) = sup Hs,δ (E) = lim Hs,δ (E), δ>0

Ss (E)

=

sup Ss,δ (E) δ>0

δ→0

= lim Ss,δ (E). δ→0

By Carath`eodory’s construction, E → Hs (E) and E → Ss (E) are Borel measures in Hn . The measure Hs is called s-dimensional Hausdorff measure. The measure Ss is called s-dimensional spherical Hausdorff measure. These measures are equivalent, in the sense that for any

74 Roberto Monti

E ⊂ Hn there holds Hs (E) ≤ Ss (E) ≤ 2s Hs (E). The measures HQ (E) and SQ are Haar measures in Hn and therefore they coincide with the Lebesgue measure, up to a multiplicative constant factor. The natural dimension to measure hypersurfaces, as the boundary of smooth sets, is s = Q − 1. The following theorem is proved in [27], Theorem 7.1 part (iii). The proof relies on Federer’s differentiation theorems, Theorem 2.10.17 and Theorem 2.10.19 part (3) of [25]. Extensions of this result are based on general differentiation theorems for measures, see [41]. Formula (2.14) for the geometric constant cn in (2.13) depends on the shape (convexity and symmetries) of the metric unit ball U1 , [42]. Theorem 2.11 (Franchi-Serapioni-Serra Cassano). For any set E ⊂ Hn with locally Fnite H -perimeter we have μ E = cn SQ−1 ∂ ∗ E,

(2.13)

where μ E is the perimeter measure of E, SQ−1 ∂ ∗ E is the restriction of SQ−1 to the reduced boundary ∂ ∗ E, and the constant cn > 0 is given by (2.14) cn = P(Hν ; U1 ). Remark 2.12. It is not known whether in (2.13) the spherical measure SQ−1 can be replaced by the Hausdorff measure HQ−1 , even when ∂ ∗ E is a smooth set. In Rn with the standard perimeter, the identity S n−1 ∂ ∗ E = H n−1 ∂ ∗ E follows from Besicovitch’s covering theorem, that fails to hold in the Heisenberg group, see [36] and [65]. 2.2.2 Minkowski content and H -perimeter In the description of H -perimeter in terms of Minkowski content, the correct choice of the metric is the Carnot-Carath´eodory distance d on Hn . The Carnot-Carath´eodory distance from a closed set K ⊂ Hn is the function dist K ( p) = min d( p, q), p ∈ Hn . q∈K

For r > 0, the r-tubular neighborhood of K is the set   Ir (K ) = p ∈ Hn : dist K ( p) < r .

75 Isoperimetric problem and minimal surfaces

The upper and lower Minkowski content of K in an open set A ⊂ Hn are, respectively, |Ir (K ) ∩ A| , 2r r→0 |Ir (K ) ∩ A| M − (K ; A) = lim inf . r→0 2r M + (K ; A) = lim sup

Above, | · | stands for Lebesgue measure. If M + (K ; A) = M − (K ; A), the common value is called Minkowski content of K in A and it is denoted by M (K ; A). Below, H 2n is the standard 2n-dimensional Hausdorff measure in n H = R2n+1 . Theorem 2.13 (Monti-Serra Cassano). Let A ⊂ Hn be an open set and let E ⊂ Hn be a bounded set with C 2 boundary such that H 2n (∂ E ∩ ∂ A) = 0. Then we have P(E; A) = M (∂ E; A).

(2.15)

This result is proved in [54], in a general framework. It is an open problem to prove formula (2.15) for sets E with less regular boundary. The tools used in the proof in [54] are the eikonal equation for the CarnotCarath´eodory distance and the coarea formula. Assume A = Hn . We have  |Ir (∂ E)| =

Ir (∂ E)

|∇H dist∂ E (z, t)|dzdt,

because |∇H dist K (z, t)| = 1 a.e. in Hn . By the coarea formula in the sub-Riemannian setting, we have 

 Ir (∂ E)

|∇H dist∂ E (z, t)|dzt =

r

P(Is (∂ E); Hn )ds.

0

We refer the reader to [54] and [40] for a discussion on coarea formulas. Now formula (2.15) follows proving that 1 lim r→0 2r



r

P(Is (∂ E); Hn )ds = P(E; Hn ).

0

The regularity of ∂ E is used at this Vnal step: the Riemannian approximation of the distance function from ∂ E is of class C 2 , if ∂ E is of class C 2 .

76 Roberto Monti

2.2.3 Integral differential quotients H -perimeter can be also expressed as the limit of certain integral differential quotients. Let kn > 0 be the following geometric constant  kn = |ν, z|dzdt, B1

where B1 ⊂ Hn is the unitary Carnot-Carath´eodory ball centered at the origin. By the rotational symmetry of B1 , the deVnition of kn is independent of the unit vector ν ∈ R2n , |ν| = 1. The following theorem is proved in [62]. Theorem 2.14. A Borel set E ⊂ Hn with Fnite measure has Fnite H perimeter in Hn if and only if   1 lim inf |χ E ( p) − χ E (q)| dp dq < ∞. r↓0 r Hn Br (q) Moreover, if E has also Fnite Euclidean perimeter then   1 lim |χ E ( p) − χ E (q)| dp dq = kn P(E; Hn ). r↓0 r Hn Br (q)

(2.16)

For the proof, we refer to [62], where the result is proved in the setting of BVH functions. It is an open question whether the identity (2.16) holds dropping the assumption “if E has also Vnite Euclidean perimeter”. The characterization of H -perimeter in Theorem 2.14 is useful in the theory of rearrangements in the Heisenberg group proposed in [49]. 2.3 RectiHability of the reduced boundary The reduced boundary of sets with Vnite H -perimeter needs not be rectiVable in the standard sense. However, it is rectiVable in an intrinsic sense that we are going to explain. The main reference is the paper [27]. A systematic treatment of these topics in the setting of stratiVed groups can be found in [39]. We need Vrst the notion of C H1 -regular function. DeHnition 2.15 (C H1 -function). Let A ⊂ Hn be an open set. A function f : A → R is of class C H1 (A) if: 1) f ∈ C(A); 2) the derivatives X 1 f, . . . , X n f, Y1 f, . . . , Yn f in the sense of distributions are (represented by) continuous functions in A. The horizontal gradient of a function f ∈ C H1 (A) is the vector valued  mapping ∇H f ∈ C(A; R2n ), ∇H f = X 1 f, . . . , X n f, Y1 f, . . . , Yn f .

77 Isoperimetric problem and minimal surfaces

For C H1 -regular functions there is an implicit function theorem (Theorem 6.5 in [27]) that can be used to represent the zero set { f = 0} as an “intrinsic Lipschitz graph” (see Section 3.1.4). DeHnition 2.16 (H -regular hypersurface). A set S ⊂ Hn is an H -regular hypersurface if for all p ∈ S there exists r > 0 and a function f ∈ C H1 (Br ( p)) such that:   1) S ∩ Br ( p) = q ∈ Br ( p) : f (q) = 0 ; 2) |∇H f ( p)|  = 0. If S ⊂ Hn is a hypersurface of class C 1 in the standard sense, then for any p ∈ S there f ∈ C 1 (Br ( p)) such that  exist r > 0 and a function  S ∩ Br ( p) = q ∈ Br ( p) : f (q) = 0 and |∇ f ( p)|  = 0. However, the set S needs not be an H -regular hypersurface because it may happen that |∇H f ( p)| = 0 at some (many) points p ∈ S. On the other hand, the following theorem, proved in [35] Theorem 3.1, shows that, in general, H -regular hypersurfaces are not rectiVable. Theorem 2.17 (Kirchheim-Serra Cassano). There exists an H -regular surface S ⊂ H1 such that H

(5−)/2

(S) > 0 for all  ∈ (0, 1).

In particular, the set S is not 2-rectiFable. Above, H s is the standard s-dimensional Hausdorff measure in R . The set S constructed in [35] has Euclidean Hausdorff dimension 5/2. Any H -regular surface S ⊂ H1 can be locally parameterized by a 1/2-H¨older continuous map  : R2 → (R2 ) = S ⊂ H1 , i.e., d((u), (v)) ≤ C|u − v|1/2 for u, v ∈ R2 , where C > 0 is a constant and d is the Carnot-Carath´eodory distance, see Theorem 4.1 in [35]. 3

DeHnition 2.18. A set  ⊂ Hn is SQ−1 -rectiVable if there exists a sequence of H -regular hypersurfaces (S j ) j∈N in Hn such that

S j = 0. SQ−1  \ j∈N

This deVnition is generalized in [43], where the authors study the notion of a s-rectiVable set in Hn for any integer 1 ≤ s ≤ Q − 1. The deVnition of s-rectiVability has a different nature according to whether s ≤ n or s ≥ n + 1. DeVnition 2.18 is relevant because the reduced boundary of sets with Vnite H -perimeter is rectiVable precisely in this sense. The following theorem is the main result of [27].

78 Roberto Monti

Theorem 2.19. Let E ⊂ Hn be a set with locally Fnite H -perimeter. Then the reduced boundary ∂ ∗ E is SQ−1 -rectiFable. The proof of Theorem 2.19 goes as follows, for details see Theorem 7.1 in [27]. By Lusin’s theorem there are compact sets K j ⊂ ∂ ∗ E, j ∈ N, and a set N ⊂ ∂ ∗ E such that: i) μ E (N ) = 0; ii) ν E : K j → R2n is continuous, for each j ∈ N; iii) ∂ ∗ E = N ∪ K j. j∈N

By a Whitney extension theorem (Theorem 6.8 in [27]), it is possible to construct functions f j ∈ C H1 (Hn ) such that ∇H f j = ν E and f j = 0 on K j . Then the sets S j = { f j = 0} are H -regular hypersurfaces near K j and K j ⊂ S j .

3 Area formulas, Hrst variation and H -minimal surfaces 3.1 Area formulas In this section, we derive some area formulas for H -perimeter of sets with regular boundary. In particular, we study sets with Euclidean Lipschitz boundary and sets with “intrinsic Lipschitz boundary”. Let E ⊂ Hn be a set with Lipschitz boundary and denote by N the Euclidean outer unit normal to ∂ E. This vector is deVned at H 2n -a.e. point of ∂ E. Here and hereafter, H 2n denotes the standard 2n-dimensional Hausdorff measure of R2n+1 . Using the projections of X 1 , . . . , X n , Y1 , . . . , Yn along the normal N , we can deVne the 2n-dimensional vector Veld N H : ∂ E → R2n   N H = X 1 , N , . . . , X n , N , Y1 , N  . . . , Yn , N  , (3.1) 3.1.1 Sets with Lipschitz boundary

where the vector Velds X j , Y j and N are identiVed with elements of R2n+1 and ·, · is the standard scalar product. Proposition 3.1. Let E ⊂ Hn be a set with Lipschitz boundary. Then the H -perimeter of E in an open set A ⊂ Hn is  |N H |dH 2n , (3.2) P(E; A) = ∂ E∩A

where N is the Euclidean (outer) unit normal to ∂ E and |N H | is the Euclidean norm of N H .

79 Isoperimetric problem and minimal surfaces

 Proof. For any ϕ ∈ Cc1 (A; R2n ) let V = nj=1 ϕ j X j + ϕn+ j Y j be the horizontal vector Veld with coordinates ϕ. By the standard divergence theorem and the Cauchy-Schwarz inequality, we have    divH ϕ dzdt = divV dzdt = V, N dH 2n E E ∂E   n = ϕ j X j , N  + ϕn+ j Y j , N dH 2n  ≤

∂ E j=1 n 

∂ E j=1

|ϕ||N H |dH

2n

,

and taking the supremum with ϕ∞ ≤ 1 it follows that P(E; A) ≤  |N H |dH 2n . ∂ E∩A

The opposite inequality can be obtained by approximation. By Lusin’s theorem, for any  > 0 there exists a compact set K ⊂ ∂ E ∩ A such that  |N H |dH 2n < , (∂ E\K )∩A

and N H : K → R2n is continuous and nonzero. By Tietze’s theorem, there exists ψ ∈ Cc (A; R2n ) such that ψ∞ ≤ 1 and ψ = N H /|N H | on K . By molliVcation there exists ϕ ∈ Cc1 (A; R2n ) such that ϕ∞ ≤ 1 and ψ − ϕ∞ < . For such a test function ϕ we have   divH ϕ dzdt ≥ (1 − ) |N H |dH 2n − 2. ∂ E∩A

E

This ends the proof. Let E ⊂ Hn be a set From the Gausswith Lipschitz boundary and let ϕ ∈ Green formula (2.6) and from the standard divergence theorem, we have    2n ϕ, N H dH = divH ϕ dzdt = − ϕ, ν E dμ E .

3.1.2 Formulas for the horizontal inner normal

Cc1 (Hn ; R2n ).

∂E

Hn

E

It follows that the perimeter measure has the following representation μ E = |N H |H

2n

∂ E,

and the measure theoretic inner normal is νE = −

NH |N H |

μ E -a.e. on ∂ E.

(3.3)

80 Roberto Monti

Next, we express ν E in terms of a deVning function for the boundary. Assume that ∂ E is a C 1 -surface and f ∈ C 1 (A) is a deVning function for ∂ E, i.e., ∂ E ∩ A = { p ∈ A : f ( p) = 0} with |∇ f |  = 0 and f < 0 inside E. Then the outer Euclidean normal to ∂ E is N=

∇f |∇ f |

on ∂ E ∩ A,

and therefore the vector N H introduced in (3.1) is NH =

∇H f |∇ f |

on ∂ E ∩ A.

From (3.3), we conclude that the horizontal inner normal is given by νE = −

∇H f |∇H f |

on ∂ E ∩ A, |∇H f |  = 0.

(3.4)

Let N E be the horizontal vector with coordinates ν E in the basis X 1 , . . . , X n , Y1 , . . . , Yn . The vector N E can be recovered in the following way. Fix on Hn the Riemannian metric making X 1 , . . . , X n , Y1 , . . . , Yn , T orthonormal. The Riemannian exterior normal to the surface { f = 0} is the vector ∇R f . NR = |∇R f | R n where ∇R f = j=1 (X j f )X j + (Y j f )Y j + (T f )T is the gradient of f and |∇R f | R is its Riemannian length. Let π R : T p Hn → H p be the orthogonal projection onto the horizontal plane. Then the vector N E is precisely π R (N R ) NE = . |π R (N R )| R We specialize formula (3.2) to the case n C be an open set and let f : D → R be of t-graphs. Let D ⊂ R2n =  n a function. The set E f = (z, t) ∈  H : t >n f (z), z ∈ D} is called t-epigraph of f . The set gr( f ) = (z, t) ∈ H : t = f (z), z ∈ D} is called t-graph of f . 3.1.3 Area formula for t-graphs

Proposition 3.2 (Area formula for t-graphs). Let D ⊂ R2n be an open set and let f : D → R be a Lipschitz function. Then we have  |∇ f (z) + 2z ⊥ |dz, (3.5) P(E f ; D × R) = D

where z ⊥ = (x, y)⊥ = (−y, x).

81 Isoperimetric problem and minimal surfaces

Proof. The outer normal to ∂ E f ∩ (D × R) = gr( f ) is N = (∇ f, −1)/  1 + |∇ f |2 , and so, for any j = 1, . . . , n, we have ∂x j f − 2y j N, X j  =  , 1 + |∇ f |2 and thus

∂ y j f + 2y j N, Y j  =  , 1 + |∇ f |2

|∇ f + 2z ⊥ | . |N H | =  1 + |∇ f |2

By formula (3.2) and by the standard area formula for graphs, we obtain   2n |N H |dH = |∇ f (z) + 2z ⊥ |dz P(E f ; D × R) = gr( f )

D

The area formula (3.5) is the starting point of many investigations on H -minimal surfaces. Epigraphs of the form E f = {t > f (z)} are systematically studied in [71]. In particular, in Theorem 3.2 of [71] the authors compute the relaxed functional in L 1 (D) of the area functional A : C 1 (D) → [0, ∞]  |∇ f (z) + 2z ⊥ |dz. A (f) = D

They also prove existence of minimizers with a trace constraint when D is a bounded open set with Lipschitz boundary (Theorem 1.4) and they show that minimizers are locally bounded (Theorem 1.5). The Lipschitz regularity of minimizers under the bounded slope condition is proved in [63]. Let S ⊂ Hn be a C H1 -regular hypersurface. Then we have S = { f = 0} with f ∈ C H1 satisfying |∇H f |  = 0. Up to a change of coordinates, we can assume that locally we have X 1 f > 0. Then each integral line of X 1 meets S in one single point: S is a graph along X 1 . These considerations lead to the following deVnitions. The line Wow of the vector Veld X 1 starting from the point (z, t) ∈ Hn is exp(s X 1 )(z, t) = (z + se1 , t + 2y1 s), s ∈ R, 3.1.4 Area formula for intrinsic graphs

where e1 = (1, 0, . . . , 0) ∈ R2n and z = (x, y) ∈ Cn = R2n , with x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ). We Vx a domain of initial points. The most natural choice is to consider   the vertical hyperplane W = (z, t) ∈ Hn : x1 = 0 , that is identiVed with R2n with the coordinates w = (x2 , . . . , xn , y1 , . . . , yn , t).

82 Roberto Monti

DeHnition 3.3 (Intrinsic epigraph and graph). Let D ⊂ W be a set and let ϕ : D → R be a function. The set   E ϕ = exp(s X 1 )(w) ∈ Hn : s > ϕ(w), w ∈ D is called intrinsic epigraph of ϕ along X 1 . The set   gr(ϕ) = exp(ϕ(w)X 1 )(w) ∈ Hn : w ∈ D

(3.6)

is called intrinsic graph of ϕ along X 1 . In DeVnition 3.8, there is an equivalent point of view on intrinsic graphs. We are going to introduce a nonlinear gradient for functions ϕ : D → R. First, let us introduce the Burgers’ operator B : Liploc (D) → L ∞ loc (D) Bϕ =

∂ϕ ∂ϕ − 4ϕ . ∂ y1 ∂t

(3.7)

Next, notice that the vector Velds X 2 , . . . , X n , Y2 , . . . , Yn can be naturally restricted to W . DeHnition 3.4 (Intrinsic gradient). The intrinsic gradient of a function 2n−1 ϕ ∈ Liploc (D) is the vector valued mapping ∇ ϕ ϕ ∈ L ∞ ) loc (D; R  ∇ ϕ ϕ = X 2 ϕ, . . . , X n ϕ, Bϕ, Y2 ϕ, . . . , Yn ϕ). When n = 1, the deVnition reduces to ∇ ϕ ϕ = Bϕ. With abuse of notation, we deVne the cylinder over D ⊂ W along X 1 as the set   D · R = exp(s X 1 )(w) ∈ Hn : w ∈ D and s ∈ R . When D ⊂ W is open, the cylinder D · R is an open set in Hn . The general version of the following proposition is presented in Theorem 3.9. Proposition 3.5. Let D ⊂ W be an open set and let ϕ : D → R be a Lipschitz function. Then the H -perimeter of the intrinsic epigraph E ϕ in the cylinder D · R is   P(E ϕ ; D · R) = 1 + |∇ ϕ ϕ|2 dw, (3.8) D

where dw is the Lebesgue measure in R2n .

83 Isoperimetric problem and minimal surfaces

Proof. We prove the claim in the case n = 1. The intrinsic graph mapping  : D → H1 is (y, t) = exp(ϕ(y, t)X)(0, y, t) = (ϕ, y, t +2yϕ), and thus     e1 e2 e3      y ∧ t =  ϕ y 1 2ϕ + 2ϕ y  = 1 + 2yϕt )e1 + 2ϕϕt − ϕ y )e2 − ϕt e3 .  ϕt 0 1 + 2yϕt  The Euclidean outer normal to the intrinsic graph ∂ E ϕ ∩ (D · R) is the vector N = − y ∧ t /| y ∧ t | and thus N, X =

−1 | y ∧ t |

and N, Y  =

ϕ y − 4ϕϕt Bϕ = . | y ∧ t | | y ∧ t |

From formula (3.2) and from the standard area formula for graphs, we obtain  P(E ϕ ; D · R) = |N H |dH2 ∂ E ϕ ∩D·R

 ! = D

1 (Bϕ)2 + | y ∧ t |dydt | y ∧ t |2 | y ∧ t |2

  1 + |∇ ϕ ϕ|2 dydt. = D

The area formula (3.8) was originally proved for boundaries that are C H1 regular hypersurfaces (see [27] Theorem 6.5 part (vi) and [4] Proposition 2.22). It was later generalized to intrinsic Lipschitz graphs. DeHnition 3.6. Let D ⊂ W = R2n be an open set and let ϕ ∈ C(D) be a continuous function. i) We say that Bϕ exists in the sense of distributions and is represented by a locally bounded function, Bϕ ∈ L ∞ loc (D), if there exists a function 1 (D) such that for all ϑ ∈ C (D) there holds ψ ∈ L∞ c loc  ϑ ψ dw = − D

 ∂ϑ  ∂ϑ − 2ϕ 2 ϕ dw. ∂ y1 ∂t D

2n−1 ) exists in ii) We say that the intrinsic gradient ∇ ϕ ϕ ∈ L ∞ loc (D; R the sense of distributions if X 1 ϕ, . . . , X n ϕ, Bϕ, Y2 ϕ, . . . , Yn ϕ are represented by locally bounded functions in D.

84 Roberto Monti

We introduce intrinsic Lipschitz graphs along any direction. Theorem 3.9 below relates such graphs to the boundedness of the intrinsic gradient ∇ ϕ ϕ. Let ν ∈ R2n , |ν| = 1, be a unit vector that is identiVed with (ν, 0) ∈ Hn . For any p ∈ Hn , we let ν( p) =  p, νν ∈ Hn and we deVne ν ⊥ ( p) ∈ ∂ Hν ⊂ Hn as the unique point such that p = ν ⊥ ( p) · ν( p).

(3.9)

Recall that  · ∞ is the box-norm introduced in (1.11). DeHnition 3.7 (Intrinsic cones). i) The (open) cone with vertex 0 ∈ Hn , axis ν ∈ R2n , |ν| = 1, and aperture α ∈ (0, ∞] is the set   (3.10) C(0, ν, α) = p ∈ Hn : ν ⊥ ( p)∞ < αν( p)∞ . ii) The cone with vertex p ∈ Hn , axis ν ∈ R2n , and aperture α ∈ (0, ∞] is the set C( p, ν, α) = p · C(0, ν, α). DeHnition 3.8 (Intrinsic Lipschitz graphs). Let D ⊂ ∂ Hν be a set and let ϕ : D → R be a function. i) The intrinsic graph of ϕ is the set   gr(ϕ) = p · ϕ( p)ν ∈ Hn : p ∈ D .

(3.11)

ii) The function ϕ is L-intrinsic Lipschitz if there exists L ≥ 0 such that for any p ∈ gr(ϕ) there holds gr(ϕ) ∩ C( p, ν, 1/L) = ∅.

(3.12)

When ν = e1 , the deVnition in (3.11) reduces to the deVnition in (3.6). Namely, let ϕ : D → R be a function with D ⊂ W = {x1 = 0}. For any w ∈ D, we have the identity exp(ϕ(w)X 1 )(w) = w · (ϕ(w)e1 ), where ϕ(w)e1 = (ϕ(w), 0 . . . , 0) ∈ Hn . Then the intrinsic graph of ϕ is the set   gr(ϕ) = w · (ϕ(w)e1 ) ∈ Hn : w ∈ D . The notion of intrinsic Lipschitz function of DeVnition 3.8 is introduced in [30]. The cones (3.10) are relevant in the theory of H -convex sets [5]. The following theorem is the Vnal result of many contributions.

85 Isoperimetric problem and minimal surfaces

Theorem 3.9. Let ν = e1 , D ⊂ ∂ Hν be an open set, and ϕ : D → R be a continuous function. The following statements are equivalent: 2n−1 ). A) We have ∇ ϕ ϕ ∈ L ∞ loc (D; R

B) For any D   D, the function ϕ : D  → R is intrinsic Lipschitz. Moreover, if A) or B) holds then the intrinsic epigraph E ϕ ⊂ Hn has locally Fnite H -perimeter in the cylinder D ·R, the inner horizontal normal to ∂ E ϕ is 1 −∇ ϕ ϕ(w)

ν Eϕ (w · ϕ(w)) =  , , 1 + |∇ ϕ ϕ(w)|2 1 + |∇ ϕ ϕ(w)|2

(3.13)

for L -a.e. w on D, 2n

and, for any D  ⊂ D, we have    P(E ϕ ; D · R) = 1 + |∇ ϕ ϕ|2 dw = cn SQ−1 (gr(ϕ) ∩ D  · R). D

(3.14)

The equivalence between A) and B) is a deep result that is proved in [7], Theorem 1.1. Formula (3.13) for the normal and the area formula (3.14) are proved in [16] Corollary 4.2 and Corollary 4.3, respectively. A related result can be found in [56], where it is proved that if E ⊂ Hn is a set with Vnite H -perimeter having controlled normal ν E , say ν E , e1  ≥ k > 0 μ E -a.e., then the reduced boundary ∂ ∗ E is an intrinsic Lipschitz graph along X 1 . 3.2 First variation and H -minimal surfaces In this section, we deduce the minimal surface equation for H -minimal surfaces in the special but important case of t-graphs. We show that H minimal surfaces in H1 are ruled surfaces. These facts have been observed by several authors. In Section 3.2.2, we review some results established in [12] and [14] about the characteristic set of surfaces in H1 with “controlled curvature”, see Theorem 3.15 below. Let D ⊂ R2n be an open set and let f ∈ C (D) be a function. Assume that the t-epigraph of f , the set   E = (z, t) ∈ Hn : t > f (z), z ∈ D , 3.2.1 First variation of the area for t-graphs 2

is H -perimeter minimizing in the cylinder A = D × R. This means that if F ⊂ Hn is a set such that EF  A then P(E; A) ≤ P(F; A). Here

86 Roberto Monti

and in the following, EF = E \ F ∪ F \ E denotes the symmetric difference of sets. Let ( f ) = {z ∈ D : ∇ f (z) + 2z ⊥ = 0} be the characteristic set of f . At points p = (z, f (z)) ∈ ∂ E with z ∈ ( f ) we have T p ∂ E = H p , the horizontal plane and the tangent plane to ∂ E at p coincide. These points are called characteristic points of the surface S = ∂ E. The set of characteristic points of S is denoted by (S). By the area formula (3.5), we have   ⊥ |∇ f (z) + 2z |dz = |∇ f + 2z ⊥ |dz. P(E; A) = D\( f )

D

By the minimality of E, for any  ∈ R and ϕ ∈ Cc∞ (D) we have   ⊥ |∇ f + 2z |dz ≤ |∇ f + ∇ϕ + 2z ⊥ |dz D\( f ) D  |∇ f + ∇ϕ + 2z ⊥ |dz + || = D\( f )  |∇ϕ|dz = ψ(). × ( f )

If f ∈ C 2 then ( f ) is (contained in) a C 1 hypersurface of D, see Section 3.2.2, and therefore |( f )| = 0. If we only have f ∈ C 1 , this is no longer true. When |( f )| = 0, the function ψ is differentiable at  = 0 and the minimality of E implies ψ  (0) = 0. We deduce that for any test function ϕ we have  ∇ f + 2z ⊥ , ∇ϕ dz = 0. |∇ f + 2z ⊥ | D\( f ) If ϕ ∈ Cc1 (D \ ( f )), we can integrate by parts with no boundary contribution obtaining 

∇ f + 2z ⊥

div ϕdz = 0. |∇ f + 2z ⊥ | D\( f )

(3.15)

When the support of ϕ intersects ( f ), there is a contribution to the Vrst variation due to the characteristic set, see Theorem 3.17. From (3.15), we deduce that the function f satisVes the following partial differential equation ∇ f + 2z ⊥

= 0 in D \ ( f ). (3.16) div |∇ f + 2z ⊥ |

87 Isoperimetric problem and minimal surfaces

This is the H -minimal surface equation for f , in the case of t-graphs. It is a degenerate elliptic equation. A solution f ∈ C 2 (D) to (3.16) is calibrated and the epigraph of f is H -perimeter minimizing over the cylinder D \ ( f ) × R DeHnition 3.10 (H -curvature and H -minimal graphs). For any f ∈ C 2 (D) and z ∈ D \ ( f ), the number ∇ f (z) + 2z ⊥

, H (z) = div |∇ f (z) + 2z ⊥ | is called H -curvature of the graph of f at the point (z, f (z)). If H = 0 we say that gr( f ) is an H -minimal graph (surface). We specialize the analysis to the dimension n = 1, where the minimal surface equation (3.16) has a clear geometric meaning. If n = 1, then ∂ E ∩ (D × R) = gr( f ) is a 2-dimensional surface. At noncharacteristic points p = (z, f (z)) ∈ ∂ E with z ∈ D \ ( f ), we have dim(T p ∂ E ∩ H p ) = 1. A section of T p ∂ E ∩ H p is the vector Veld   1 + 2x)X + ( f − 2y)Y . − ( f V = y x |∇ f + 2z ⊥ | Let γ : (−δ, δ) → H1 , δ > 0, be the curve such that γ (0) = p ∈ ∂ E and γ˙ = V (γ ). The curve γ is horizontal because V is horizontal. Moreover, we have γ (t) ∈ ∂ E for all t ∈ (−δ, δ) because V is tangent to ∂ E. Consider the vector Velds in D \ ( f )   − f y − 2x, f x − 2y ∇ f + 2z ⊥ ⊥ and N f (z) = . N f (z) = |∇ f + 2z ⊥ | |∇ f + 2z ⊥ | The vector Veld N ⊥f is the projection of V onto the x y-plane. The horizontal projection of γ , the curve κ = (γ1 , γ2 ), satisVes κ(0) = z 0 and solves the differential equation κ˙ = N ⊥f (κ). Then the vector N f is a normal vector to the curve κ. Viceversa, let κ be the solution of κ˙ = N ⊥f (κ) and κ(0) = z 0 and let γ be the horizontal lift of κ with γ (0) = p = (z 0 , f (z 0 )) ∈ ∂ E. Then γ solves γ˙ = V (γ ) and is contained in ∂ E. We summarize these observations in the following proposition. Proposition 3.11. Let S = gr( f ) ⊂ H1 be the graph of a function f ∈ C 1 (D). Then: 1) The horizontal projection κ of a horizontal curve γ contained in S \ (S) solves κ˙ = N ⊥f (κ).

88 Roberto Monti

2) The horizontal lift γ of a curve κ solving κ˙ = N ⊥f (κ) in D \ ( f ) is contained in S, if γ starts from S. Now it is straightforward to prove the following result. Theorem 3.12 (Structure of H -minimal surfaces). Let D ⊂ C be an open set and let f ∈ C 2 (D) be a function such that gr( f ) is an H minimal surface. Then for any z 0 ∈ D \ ( f ) there exists a horizontal line segment contained in gr( f ) and passing through (z 0 , f (z 0 )). Proof. Let γ be the horizontal curve passing through p = (z 0 , f (z 0 )) and contained in gr( f ). The horizontal projection κ solves κ˙ = N f (κ)⊥ . The minimal surface equation (3.16) reads div N f (z) = 0

in D \ ( f ),

where N f is a unit normal vector Veld of κ. Thus κ is a curve with curvature 0 and thus it is a line segment. Its horizontal lift is also a line segment. Remark 3.13. If H : D \ ( f ) → R is the H -curvature of the graph of f , then the partial differential equation ∇ f (z) + 2z ⊥

= H (z), div |∇ f (z) + 2z ⊥ |

in D \ ( f ) ⊂ C,

implies that an integral curve κ of the vector Veld N ⊥f has curvature H (κ). When H is a nonzero constant, κ is a circle. This is relevant in the Heisenberg isoperimetric problem. Equation (3.16) can be given a meaning along integral curves of N ⊥f without assuming the full C 2 regularity of f , see [13]. See also Section 4.3 for the problem of integrating the H -curvature equation for a convex function f . Let D ⊂ C2n be an open set and let f ∈ C (D). Consider the mapping  : D → R2n 3.2.2 Characteristic points 2

(z) = ∇ f (z) + 2z ⊥ ,

z ∈ D.

The point z = x + i y ∈ ( f ) is characteristic if and only if (z) = 0, namely,  1 (z) = ∇x f (z) − 2y = 0 2 (z) = ∇ y f (z) + 2x = 0.

89 Isoperimetric problem and minimal surfaces

If z 0 ∈ ( f ) is a point such that det(J (z 0 ))  = 0 then  is a local C 1 diffeomorphism at z 0 and thus z 0 is an isolated point of ( f ). In general, for any z 0 ∈ ( f ) there exists  > 0 such that ( f ) ∩ {|z − z 0 | < } is contained in the graph of a C 1 function. For instance, in the case n = 1 we have |∂ y 1 (z)| + |∂x 2 (z)| = | f x y (z) − 2| + | f x y (z) + 2|  = 0, and the claim follows from the implicit function theorem. We used the C 2 regularity of f to have equality of mixed derivatives f x y = f yx . When f is less than C 2 -regular, the characteristic set ( f ) may be large. 2 Theorem 3.14 (Balogh). Let D = (0, 1) × (0, 1) "⊂ R be the square. C 1,α (D) such that For any  > 0 there exists a function f ∈ 0 0.

C , |z − z 0 |

z ∈ D \ ( f )

(3.18)

90 Roberto Monti

Then there exists  > 0 such that ( f ) ∩ {|z − z 0 | < } is the graph of a C 1 function deFned over an open interval. Proof. Since det(J (z 0 )) = 0 then the Jacobian matrix J (z 0 ) has rank at most 1. On the other hand, the antidiagonal of J (z 0 ) never vanishes and thus the rank is precisely 1. Up to the sign, there exists a unique unit vector w ∈ R2 , |w| = 1, that is orthogonal to the range of the transposed Jacobian matrix J (z 0 )∗ . For u ∈ R2 , we deVne the function u : D → R, u = , u = / Ker(J (z 0 )∗ ) then u 1 ( f x − 2y) + u 2 ( f y + 2x). If u ∈ ∇u (z 0 ) = J (z 0 )∗ u = 0, and thus the equation u = 0 deVnes a C 1 curve κu : (−s0 , s0 ) → R2 , for some s0 > 0, such that κu (0) = z 0 and u (κu ) = 0. The image of this curve is a graph over an interval. We can assume that |κ˙ u | = 1. Differentiating u (κu ) = 0 we obtain ∇u (κu ), κ˙ u  = 0, and therefore at s = 0 we have J (z 0 )∗ u, κ˙ u (0) = 0. Then, up to the sign we have κ˙ u (0) = w. The derivative κ˙ u (0) is independent of u ∈ / Ker(J (z 0 )∗ ). For some small  > 0, we have ( f ) ∩ {|z − z 0 | < } ⊂ {κu (s) ∈ R2 : |s| < s0 }∩{|z−z 0 | < }. We claim that the inclusion is an identity of sets. By contradiction assume that for any δ > 0 there are 0 ≤ s1 < s2 ≤ δ / ( f ) for s1 < s < s2 , and κu (s1 ), κu (s2 ) ∈ ( f ). such that κu (s) ∈ Without loss of generality, we assume that s1 = 0 and s2 = δ, where δ > 0 is as small as we wish. The deVning equation (κu ), u = u (κu ) = 0 implies that, for 0 < s < δ, the vector N f (κu (s)) =

(κu (s)) = ±u ⊥ |(κu (s))|

(3.19)

is constant, either +u ⊥ or −u ⊥ , where u ⊥ = (−u 2 , u 1 ). / Ker(J (z 0 )∗ ), There exists a unit vector v ∈ R2 such that v ∈ u − v, w = 0 and u + v, w  = 0.

(3.20)

The equation v = 0 deVnes a C 1 curve κv : (−¯s0 , s¯0 ) → R2 such that κv (0) = z 0 , κ˙ v (0) = w, |κ˙ v | = 1 and v (κv ) = 0. There is a number δ¯ > ¯ = κu (δ) and κv (s) ∈ ¯ As above, / ( f ) for 0 < s < δ. 0 such that κv (δ) ¯ the the equation (κv ), v = v (κv ) = 0 implies that, for 0 < s < δ, vector N f (κv (s)) = ±v ⊥ is constant.

91 Isoperimetric problem and minimal surfaces

Let A ⊂ R2 be the region enclosed by the curves κu restricted to [0, δ] ¯ Integrating the equation (3.17) over A, using and κv restricted to [0, δ]. the divergence theorem and (3.18), we obtain   N f , N dH 1 = div N f (z) dz ∂A A   (3.21) 1 = H (z)dz ≤ C dz, A A |z − z 0 | where N is the exterior normal to ∂ A. Namely, along κu we have N = κ˙ u⊥ and along κv we have N = −κ˙ v⊥ , or viceversa. Using (3.19), we can compute the integral  δ  1 N f , N dH = N f (κu (s)), κ˙ u⊥ (s)ds κu ([0,δ])

0

= ±u ⊥ , κu (δ)⊥ − z 0⊥  = ±u, κu (δ) − z 0 , ¯ = where κu (δ) − z 0 = δw + o(δ) as δ → 0. Analogously, using κv (δ) κu (δ) we obtain  δ¯  N f , N dH 1 = − N f (κv (s)), κ˙ v⊥ (s)ds = −±v, κu (δ)−z 0 , ¯ κv ([0,δ])

0

and, therefore, by (3.20) we have for δ > 0 small     δ   N f , N dH 1  ≥ u ± v, δw + o(δ) ≥ |u ± v, w|.  2 ∂A

(3.22)

Fix  > 0. For δ > 0 small, we  have the inclusion A ⊂  a parameter z 0 + rweiϑ ∈ C : 0 ≤ r ≤ δ, |ϑ| ≤  . Using polar coordinates centered at z 0 , we Vnd  1 dz ≤ 2δ, (3.23) A |z − z 0 | and, from (3.21)-(3.22)-(3.23), we obtain 2δ |u ± v, w| ≤ 2δC, that is a contradiction if we choose  > 0 such that 4C < |u ± v, w|. Let D ⊂ C be an open set, f ∈ C 2 (D), and assume that ( f ) is a C 1 curve disconnecting D. Then we have the partition D = D + ∪ D − ∪ ( f ) where D + , D − ⊂ D are disjoint open sets. In [14], Proposition 3.5, it is shown that the vector N f extends to ( f ) from D + and from D − , separately.

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Theorem 3.16. In the above setting, for any z 0 ∈ ( f ) the following limits do exist N f (z 0 )+ = z→z lim N f (z), 0

z∈D +

N f (z 0 )− = z→z lim N f (z), 0

z∈D −

and moreover N f (z 0 )+ = −N f (z 0 )− . Proof. Without loss of generality, we assume that z 0 = 0. We have either f x y (0) − 2  = 0 or f x y (0) + 2 = 0. Assume that f x y (0) − 2 > 0. equation for ( f ) near 0 and ( f ) = Then f x − 2y = 0 is a deVning   (x, ϕ(x)) ∈ R2 : |x| < δ , where ϕ ∈ C 1 (−δ, δ) is such that ϕ(0) = 0, and     D + = (x, y) ∈ D : y > ϕ(x) = z ∈ D : f x (z) − 2y > 0 ,     D − = (x, y) ∈ D : y < ϕ(x) = z ∈ D : f x (z) − 2y < 0 . By Cauchy theorem, for any x ∈ (−δ, δ) and for any y > ϕ(x) there exists ϕ(x) ¯ ∈ (ϕ(x), y) such that f yy (x, ϕ(x)) ¯ f y (x, y) + 2x = . f x (x, y) − 2y f x y (x, ϕ(x)) ¯ −2 When x → 0 and y → 0 we also have ϕ(x) ¯ → 0. Then we have lim

z→0 z∈D +

f yy (0) f y (z) + 2x = = b. f x (z) − 2y f x y (0) − 2

Using f x (z) − 2y > 0 on D + , it follows that N f (0)+ = lim N f (z) = lim z→0 z∈D +

z→0 z∈D +

∇ f (z) + 2z ⊥ (1, b) . =√ ⊥ |∇ f (z) + 2z | 1 + b2

An analogous computation using f x (z) − 2y < 0 on D − shows that N f (0)− = lim N f (z) = lim z→0 z∈D −

z→0 z∈D −

∇ f (z) + 2z ⊥ (1, b) . = −√ |∇ f (z) + 2z ⊥ | 1 + b2

For H -minimal graphs, the vectors N +f and N −f are tangent to the C 1 curve ( f ). The following theorem and Theorem 3.16 fail when we have only f ∈ C 1,1 , see Section 5.2.2.

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Theorem 3.17. In the above setting, assume that the epigraph of f ∈ C 2 (D) is H -perimeter minimizing in the cylinder D × R. Then we have N +f , N  = N −f , N  = 0 on ( f ), where N is the normal to the C 1 curve ( f ). Proof. Let ϕ ∈ Cc1 (D) be a test function and consider the function  ψ() = |∇ f + ∇ϕ + 2z ⊥ |dz,  ∈ (−0 , 0 ). D

If the epigraph of f is H -perimeter minimizing then  ∇ f + 2z ⊥ , ∇ϕ  dz. 0 = ψ (0) = |∇ f + 2z ⊥ | D By |( f )| = 0 and by (3.16), this is equivalent to   ∇ f + 2z ⊥

∇ f + 2z ⊥

div ϕ div ϕ dz + dz = 0. |∇ f + 2z ⊥ | |∇ f + 2z ⊥ | D+ D− Denoting by N the exterior unit normal to D + along ( f ) and by N +f and N −f the traces of N f onto ( f ) from D + and D − , the divergence theorem gives   0= ϕN, N +f dH 1 − ϕN, N −f dH 1 ( f ) ( f )  =2 ϕN, N +f dH 1 . ( f )

In fact, by Theorem 3.16 we have N −f = −N +f . Since ϕ is arbitrary, we conclude that N, N +f  = 0 on ( f ). 3.2.3 First variation of the area functional for intrinsic graphs By (3.14), the H -perimeter of the intrinsic epigraph E ϕ along X 1 of an intrinsic Lipschitz function ϕ : D → R, D ⊂ Cn open set, is   A (ϕ) = P(E ϕ ; D · R) = 1 + |∇ ϕ ϕ|2 dw, (3.24) D

where ∇ ϕ ϕ is a distribution represented by L ∞ (D; R2n−1 ) functions. It is not clear how to compute the Vrst variation of the area functional A within the class of intrinsic Lipschitz functions. In fact, this class is not

94 Roberto Monti

a vector space because the Burgers’ operator is nonlinear. Even for a smooth function ψ ∈ C ∞ (D) we have B(ϕ +ψ) = ϕ y +ψ y −4(ϕ +ψ)(ϕt +ψt ) = Bϕ +Bψ −4(ϕψt +ψϕt ), and the distributional derivative ϕt is not represented by an L ∞ function. So, if ϕ is only intrinsic Lipschitz it may happen that P(E ϕ+ψ ; D · R) = ∞ for any small perturbation ψ = 0. The reason of this phenomenon is that the variation of the intrinsic graph of ϕ along X 1 is not a contact deformation. On the other hand, if we had ϕt ∈ L ∞ loc then the intrinsic graph would have the standard Lipschitz regularity. Assuming the Lipschitz regularity for ϕ, the Vrst variation for the area functional A in (3.24), namely the condition d A (ϕ + ψ) = 0 d

for any ψ ∈ Cc∞ (D),

leads to the following minimal surface equation for a minimizer ϕ in D: n

∂  X jϕ ∂

Bϕ  + Xj  − 4ϕ ∂y ∂t 1 + |∇ ϕ ϕ|2 1 + |∇ ϕ ϕ|2 j=2

Yjϕ = 0. + Yj  1 + |∇ ϕ ϕ|2

(3.25)

This equation, but in a different system of coordinates, is the starting point of the papers [10] and [9], where the authors study the regularity of vanishing viscosity Lipschitz continuous solutions. When n ≥ 2, vanishing viscosity solutions are C ∞ -smooth. When n = 1, their intrinsic graph is foliated by horizontal lines. 3.3 First variation along a contact Iow In this section, we present a formula for computing the Vrst variation of H -perimeter for any set with Vnite H -perimeter. This result can be extended to SQ−1 -rectiVable sets in the sense of DeVnition 2.18 and is a joint result with D. Vittone. We give the proof in the smooth case, the technical details for the general case will appear elsewhere. First and second order variation formulas are discussed also in [45], [18], and [31]. Let A ⊂ Hn be an open set. A diffeomorphism  : A → Hn is said to be a contact map if for any p ∈ A the differential ∗ : T p Hn → T( p) Hn maps the horizontal space H p into H( p) : ∗ (H p ) = H( p) ,

p ∈ A.

(3.26)

95 Isoperimetric problem and minimal surfaces

A one-parameter Wow (s )s∈R of diffeomorphisms in Hn is a contact Gow if each s is a contact map. Contact Wows are generated by contact vector Velds. A contact vector Veld in Hn is a vector Veld of the form n  Vψ = −4ψ T + (Y j ψ)X j − (X j ψ)Y j , (3.27) j=1

where ψ ∈ C ∞ (Hn ) is the generating function of the vector Veld (see [36]). For any compact set K ⊂ Hn , there exist δ = δ(ψ, K ) > 0 ˙ p) = Vψ ((s, p)) and a Wow  : [−δ, δ] × K → Hn deVned by (s, and (0, p) = p for any s ∈ [−δ, δ] and p ∈ K . We call  the Wow generated by ψ. We also let s = (s, ·). Related to the function ψ, we have, at any point p ∈ Hn , the real quadratic form Qψ : H p → R n n 

 Qψ x j X j + yjYj = xi x j X j Yi ψ j=1

i, j=1

+ x j yi (Yi Y j ψ − X j X i ψ) − yi y j Y j X i ψ,

(3.28)

where x j , y j ∈ R, and ψ with its derivatives are evaluated at p. In the sequel, we identify a vector ν = ν( p) ∈ R2n , p ∈ Hn , with the horizontal vector nj=1 ν j X j ( p)+νn+ j Y j ( p). The quadratic form Qψ (ν) is deVned accordingly. Theorem 3.18. Let A ⊂ Hn be an open set and let  : [−δ, δ] × A → Hn , δ = δ(ψ, A) > 0, be the Gow generated by ψ ∈ C ∞ (Hn ). Then there exists a constant C = C(ψ, A) > 0 such that for any set E ⊂ Hn with Fnite H -perimeter in A we have         P(s (E); s (A)) − P(E; A) + s 4(n + 1)T ψ + Qψ (ν E ) dμ E   A

≤ C P(E; A) s 2 (3.29)

for any s ∈ [−δ, δ].

Proof. We prove the theorem when ∂ E ∩ A is a C ∞ smooth hypersurface. We deduce formula (3.29) from the Taylor expansion for the standard perimeter. Let E s = s (E) and As = s (A). Then ∂ E s ∩ As = s (∂ E ∩ A) is a C ∞ smooth 2n-dimensional hypersurface. By the area formula (3.2), we have   2n K dH and P(E s ; As ) = K s dH 2n , P(E; A) = ∂ E∩A

∂ E s ∩As

96 Roberto Monti

where H

2n

is the standard 2n-dimensional Hausdorff measure of R2n+1 , K =

n 

1/2 X j , N 2 + Y j , N 2 , j=1

n 

1/2 X j , Ns 2 + Y j , Ns 2 , Ks = j=1

and N , Ns are the standard Euclidean unit normals to ∂ E ∩ A and ∂ E s ∩ As , respectively. We Vx a coherent orientation. By the standard Taylor formula for the area, we have 



∂ E s ∩As

K s dH

2n

=

∂ E∩A

K s ◦ s Js dH

2n

,

(3.30)

where Js : ∂ E ∩ A → R is the Jacobian determinant of s restricted to ∂ E: #  ∗   Js = det J s ∂ E ◦ J s ∂ E . (3.31) This Jacobian determinant has the following Vrst order Taylor expansion in s   Js = 1 + s div Vψ − (J Vψ )N, N  + O(s 2 ) on ∂ E ∩ A, (3.32) where div Vψ is the standard divergence of the vector Veld Vψ generating the Wow and J Vψ is the Jacobian matrix of Vψ . Here, the vector Veld Vψ is identiVed with the mapping given by the coefVcients of Vψ in the standard basis. The remainder O(s 2 ) in (3.32) satisVes |O(s 2 )| ≤ C1 s 2 for some constant C1 = C1 (ψ, A) > 0. We compute the derivative of the function s → K s ◦ s . We start from the derivative of s → M(s) = Ns (s ). Let us Vx a frame V1 , . . . , V2n of orthonormal vector Velds (in the standard scalar product) tangent to ∂ E ∩ . This frame does always exist locally. As the vector Velds J s V1 , . . . , J s V2n are tangent to ∂ E s ∩ s we can differentiate the identities J s Vi , M(s) = 0, i = 1, . . . , 2n. We obtain J Vψ (s )Vi ), M(s) + J s Vi , M  (s) = 0.

(3.33)

On the other hand, differentiating the identity |Ns |2 = 1 we deduce that M  (s), Ns (s ) = 0. Using (3.33), we deduce that at the point s = 0 we

97 Isoperimetric problem and minimal surfaces

have 2n 2n    M (0) = Vi , M (0)Vi = − (J Vψ )Vi , N Vi 

i=1

i=1

2n  =− Vi , (J Vψ )∗ N Vi

(3.34)

i=1

= (J Vψ )∗ N, N N − (J Vψ )∗ N . Using the property of Wows, we can repeat the computation for any s and we Vnd the formula M  (s) = (J Vψ )∗ Ns , Ns Ns − (J Vψ )∗ Ns ,

(3.35)

where the right-hand side is evaluated at s . Now let X be any smooth vector Veld in Hn and consider the function FX (s) = X, Ns (s ). The derivative of FX is FX (s) = (J X)Vψ (s ), M(s) + X (s ), M  (s), where J X is the Jacobian matrix of the mapping given by the coefVcients of X. We may also use the notation (J X)Vψ = Vψ X, where Vψ acts on the coefVcients of X. Using (3.35), we obtain % $ FX (s) = (J X)Vψ , Ns  + X, (J Vψ )∗ Ns , Ns Ns − (J Vψ )∗ Ns (3.36) = [Vψ , X], Ns  + (J Vψ )Ns , Ns X, Ns . The right-hand side is evaluated at s . As Vψ is of the form (3.27), the commutators [Vψ , X j ] and [Vψ , Y j ] are horizontal vector Velds, i.e., linear combinations of X i and Yi . From (3.36) it follows that FX j and FY j are homogeneous functions of degree 1 with respect to X i , Ns  and Yi , Ns , i = 1, . . . , n. As s is a contact Wow, by (3.26) we have K ( p) = 0 if and only if K s (s ( p)) = 0. Assuming that K ( p) = 0, we can thus compute the derivative (in the sequel we omit reference to p ∈ ∂ E ∩ A) n d K s ◦ s 1  X j , Ns FX j (s) + Y j , Ns FY j (s), = ds K s j=1

(3.37)

and using (3.36) we obtain the formula d K s ◦ s = K s ((J Vψ )Ns , Ns  ds n & ' 1  X j , Ns [Vψ ,X j ] + Y j ,Ns [Vψ ,Y j ],Ns . + K s j=1

(3.38)

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The right hand side is evaluated at s and it is bounded by K s . Namely, there exists a constant C2 = C2 (ψ, A) such that dK ◦   s  s   ≤ C2 K s . ds

(3.39)

Then we can interchange integral and derivative in s in the derivative of P(E s ; As ):  

d d 2n K s ◦ s Js dH = K s ◦ s Js dH 2n . ds ∂ E∩A ∂ E∩A ds A formula for the second derivative of s → K s ◦ s can be obtained starting from (3.37) and using (3.36). We do not compute this formula, here. It sufVces to notice that also the second derivative is bounded by K s , and namely:  d2 K ◦   s s  (3.40)   ≤ C3 K s ds 2 for some C3 = C3 (ψ, A) > 0. This follows again from the formula (3.36). Thus we can differentiate twice in s inside the integral (3.30) deVning P(E s ; As ). From (3.32) and (3.38), we get the Vrst order Taylor development n

 1  N X j [Vψ , X j ] K s ◦ s Js = K 1 + s div Vψ + 2 K j=1   + NY j [Vψ , Y j ], N  + O(s 2 ) ,

(3.41)

where we let N X j = X j , N  and NY j = Y j , N , and O(s 2 )/s 2 is bounded uniformly in N by some constant C4 = C4 (ψ, A) > 0. Now, using the structure (3.27) of Vψ , we get n n 

 N X j [Vψ ,X j ]+ NY j [Vψ ,Y j ], N  = −Qψ N X j X j+NY j Y j , (3.42) j=1

j=1

and div Vψ = −4T ψ +

n 

X j Y j ψ − Y j X j ψ = −4(n + 1)T ψ.

j=1

Formula (3.29) follows from (3.30) along with (3.41)–(3.43).

(3.43)

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Remark 3.19. Let  ⊂ Hn be an SQ−1 -rectiVable set in Hn in the sense of DeVnition 2.18. Using the C H1 -regular surfaces that cover , a unit horizontal normal ν can be deVned SQ−1 -a.e. on . When  is bounded and with Vnite measure, formula (3.29) reads as follows:      Q−1   Q−1 Q−1  S (s ()) − S () + s 4(n + 1)T ψ + Qψ (ν ) dS   

≤ CS Q−1 () s 2 (3.44) for any s ∈ [−δ, δ], where ψ ∈ C (H ) is a generating function and δ > 0. The details of the proof of (3.44) will appear elsewhere. If  is locally measure minimizing in an open set A ⊂ Hn , from (3.44) we deduce the necessary condition    4(n + 1)T ψ + Qψ (ν ) dS Q−1 = 0 ∞

n



for any function ψ ∈ C ∞ (A).

4 Isoperimetric problem 4.1 Existence of isoperimetric sets and Pansu’s conjecture For a measurable set E ⊂ Hn with positive and Vnite measure, the isoperimetric quotient is deVned as I (E) =

P(E; Hn ) . |E|(Q−1)/Q

The isoperimetric problem consists in minimizing the isoperimetric quotient among all admissible sets   Cisop = inf I (E) : E ⊂ Hn measurable set with 0 < |E| < ∞ . (4.1) A measurable set E ⊂ Hn with 0 < |E| < ∞ realizing the inVmum is called isoperimetric set. Isoperimetric sets are deVned up to null sets. If a set E is isoperimetric, then also the left translates L p E = p · E, p ∈ Hn , are isoperimetric because perimeter and volume are left invariant. Also the dilated sets λE = δλ E are isoperimetric, because the isoperimetric quotient is 0-homogeneous, I (λE) = I (E), for any λ > 0. It follows that the inVmum Cisop in (4.1) is the inVmum of perimeter for Vxed volume   Cisop = inf P(E; Hn ) : E ⊂ Hn measurable set with |E| = 1 . (4.2)

100 Roberto Monti

Hence, isoperimetric sets are precisely the sets that have least Heisenberg perimeter for given volume. The inVmum in (4.1) is in fact positive, Cisop > 0, and we have the isoperimetric inequality P(E; Hn ) ≥ Cisop |E|

Q−1 Q

,

(4.3)

holding for any measurable set E with Vnite measure. The constant Cisop is the largest constant making true the above inequality (i.e., the sharp constant). Isoperimetric sets are precisely the sets for which the inequality (4.3) is an equality. Inequality (4.3) with a positive nonsharp constant can be obtained by several methods (see, for example, [58], [59], [26], and [33]). The functional analytic proof casts the isoperimetric inequality as a special case of Sobolev-Poincar`e inequalities. Indeed, for any 1 ≤ p < Q there exists a constant Cn, p > 0 such that p

Q− 

1/ p  pQ pQ Q− p |u| dzdt ≤ |∇H u| p dzdt (4.4) C p,n Hn

Hn

for any u ∈ Cc1 (Hn ). The inequality extends to appropriate Sobolev or BV spaces. The case p = 1 is the geometric case and reduces to the Heisenberg isoperimetric inequality (4.3). In fact, for the characteristic function of a set u = χ E we have    |∇H u| = sup χ E divH ϕ dzdt : ϕ ∈ Cc1 (A; R2n ), ϕ∞ ≤ 1 Hn

Hn

= P(E; Hn ). Inequality (4.4) can be obtained starting from the potential estimate  |∇H u(ζ, τ )| dζ dτ |u(z, t)| ≤ Cn Q−1 n H d((z, t), (ζ, τ )) = Cn I Q−1 (|∇H u|)(z, t), u ∈ Cc1 (Hn ), and using the fact that the singular integral operator I Q−1 : L p (Hn ) → L q (Hn ) is bounded for q = pQ/(Q − p) and 1 ≤ p < Q. The existence of isoperimetric sets is established in [38] and follows from a concentration-compactness argument. See also [32] for a proof of existence that avoids to use the concavity of the isoperimetric proVle function. Theorem 4.1 (Leonardi-Rigot). Let n ≥ 1. There exists a measurable set E ⊂ Hn with |E| = 1 realizing the minimum in (4.2).

101 Isoperimetric problem and minimal surfaces

Proof. We give a sketch of the proof. Let (E j ) j∈N be a minimizing sequence of sets for (4.2): 1) |E j | = 1 for all j ∈ N; 2) lim P(E j ; Hn ) = Cisop . j→∞

The key step of the proof is a concentration argument. We claim that there exists an R > 0 such that (after a left translation, truncation, and dilation of each E j ) the sequence (E j ) j∈N can be also assumed to lie in a bounded region. Namely, there exists R > 0 such that:   3) E j ⊂ Q R = (z, t) ∈ Hn : |xi |, |yi |, |t|2 < R, i = 1, . . . , n for all j ∈ N. Then, by the compactness theorem for BVH (Q R ) functions (see [33]), there exists a subsequence, still denoted by (E j ) j∈N , that converges in L 1 (Hn ) to a set E ⊂ Hn such that: i) |E| = lim |E j | = 1, by the L 1 (Hn ) convergence; j→∞

ii) P(E; Hn ) ≤ lim inf P(E j ; Hn ) = Cisop , by the lower semicontinuity j→∞

of perimeter. So we have P(E; Hn ) = Cisop with |E| = 1, and E is therefore an isoperimetric set. This ends the proof, provided that we show 3). Claim 3) follows from the following lemma. Lemma 4.2. Let n ≥ 1. There exist constants 0 > 0, C > 0, and R > 0 such that for each 0 <  < 0 and for all sets E ⊂ Hn such that |E| = 1 and P(E; Hn ) ≤ (1 + )Cisop there exists a set F ⊂ Hn such that: i) |F| = 1;   ii) F ⊂ Q R = (z, t) ∈ Hn : |xi |, |yi |, |t|2 < R, i = 1, . . . , n ;  Q −(Q−1)/Q iii) P(F; Hn ) ≤ 1 − C Q−1 P(E; Hn ). Proof. For s ∈ R, let us deVne the following sets:     n n and + − s = (z, t) ∈ H : x 1 < s s = (z, t) ∈ H : x 1 > s .   We also let s = (z, t) ∈ Hn : x1 = s . Let E ⊂ Hn be a set with |E| = 1 and Vnite H -perimeter. We deVne the sets E s− = E ∩ − s

and

E s+ = E ∩ + s .

By the Heisenberg isoperimetric inequality (4.3), we have P(E s− ; Hn ) ≥ Cisop |E s− |

Q−1 Q

,

P(E s+ ; Hn ) ≥ Cisop |E s+ |

Q−1 Q

,

(4.5)

102 Roberto Monti

where P(E s− ; Hn ) = P(E; Hs− ) + P(E s− ; s ), P(E s+ ; Hn ) = P(E; Hs+ ) + P(E s+ ; s ).

(4.6)

The number P(E s− ; s ) is the standard 2n-dimension measure of the trace of E s− onto s . Analogously, the number P(E s+ ; s ) is the standard 2n-dimension measure of the trace of E s+ onto s . The function v(s) = |E s− | is continuous and increasing. Therefore it is differentiable almost everywhere. Hence, at differentiability points s ∈ R of v we have v  (s) = P(E s− ; s ) = P(E s+ ; s ). We do not prove these claims, here. From (4.6) and (4.5), we obtain +  P(E; Hn ) + 2v  (s) ≥ P(E; − s ) + P(E; s ) + 2v (s) + − = P(E; − s ) + P(E; s ) + P(E s ; s ) + P(E s+ ; s ) = P(E s− ; Hn ) + P(E s+ ; Hn )

 Q−1 Q−1 ≥ Cisop |E s− | Q + |E s+ | Q .

Using P(E; Hn ) ≤ Cisop (1+) and |E| = 1, the inequality above implies  Q−1 Q−1  Cisop (1 + ) + 2v  (s) ≥ Cisop v(s) Q + (1 − v(s)) Q , and letting ψ(v) = v

Q−1 Q

+(1−v)

Q−1 Q

−1 for v ∈ [0, 1], we Vnally obtain

Cisop  + 2v  (s) ≥ Cisop ψ(v(s)).

(4.7)

The function ψ is strictly concave with ψ(0) = ψ(1) = 0. Then there exist 0 < v− < v+ < 1 such that ψ(v− ) = ψ(v+ ) = 2. By concavity, we have ψ(v) ≥ 2 for all v− ≤ v ≤ v+ . There exist numbers s− < s+ such that v(s− ) = v− and v(s+ ) = v+ . Thus, from (4.7) we get  s+ Cisop  + 2v  (s) s+ − s− ≤ ds Cisop ψ(v(s)) s−  s+ 2v  (s) 1 ds ≤ (s+ − s− ) + (4.8) 2 s− C isop ψ(v(s))  1 2 1 dv. ≤ (s+ − s− ) + 2 0 C isop ψ(v)

103 Isoperimetric problem and minimal surfaces

We obtain the bound s+ − s− = 2 ≤R 2 Cisop

 0

1

1 dv < ∞. ψ(v)

 = E ∩ {(z, t) ∈ Hn : s− < x1 < s+ } has volume The set E  = |E s− | − |E s− | = 1 − 2v− . | E| + − We used the identity v+ = 1 − v− . The number 0 < v− < 1/2 satisVes ψ(v− ) = 2. There are constants 0 > 0 and C > 0 such that if 0 <  < Q  = 1. Then we have 0 we have v− ≤ C Q−1 . Let λ > 0 be such that |λ E| Q Q  Q 1 = λ | E| ≥ λ (1 − 2C Q−1 ), and thus

1/Q 1 . λ≤ Q 1 − 2C Q−1  Hn ) ≤ P(E; Hn ). We do not A calibration argument shows that P( E; prove this claim, here. So we get

(Q−1)/Q 1  Hn ) ≤  Hn ) = λ Q−1 P( E; P(E; Hn ). P(λ E; Q Q−1 1 − 2C After a left translation, we may assume that    ⊂ (z, t) ∈ Hn : |x1 | < R , λE  Repeating the argument for each coordinate axis, where we let R = λ R. we obtain the claim of the lemma. The argument in the t coordinate requires easy adaptations. In 1983, Pansu conjectured a possible solution to the Heisenberg isoperimetric problem, see [59]. The conjecture can be formulated in the following way. Up to a null set, a left translation, and a dilation, the isoperimetric set in H1 is precisely the set

  E isop = (z, t) ∈ H1 : |t| < arccos |z| + |z| 1 − |z|2 , |z| < 1 . (4.9) Pansu did not give the formula for the conjectured isoperimetric set but he described how to construct it. Let us consider a geodesic γ : [0, π] → H1 joining the point γ (0) = 0 to the point γ (π) = (0, π) ∈ H1 . Using the formula (1.14) with ϑ = 0 and ϕ = 2, we have the following formula for γ

e2is − 1 , s − sin s cos . γ (s) = 2

104 Roberto Monti

2is

The horizontal projection of γ , namely the curve κ(s) = e 2−1 , is a circle with diameter 1. Letting |z| = |κ(s)| we Vnd |z|2 = 1 − cos2 s, and when s ∈ [0, π/2] we get  s = arccos 1 − |z|2 . We can thus deVne the proVle function ϕ : [0, 1] → R by letting π ϕ(|z|) = s − sin s cos s − 2   π = arccos 1 − |z|2 − |z| 1 − |z|2 − 2  2 = − arccos |z| − |z| 1 − |z| . The proVle ϕ gives the radial value of the function whose graph is the bottom part of the boundary of the set E isop in (4.9). Pansu’s conjecture is in H1 . Of course, the formula deVning E isop in (4.9) makes sense in Hn for n ≥ 2 and the conjecture can be naturally extended to any dimension. Proposition 4.3. The set E isop ⊂ H1 has the following properties: 1) The boundary ∂ E isop is of class C 2 but not of class C 3 . 2) The set E isop is convex. 3) The set E isop is axially symmetric. Proof. 1) The boundary ∂ E isop is of class C ∞ away from the center of the group Z = {(0, t) ∈ H1 : t ∈ R}. We claim that the function ϕ : [0, 1] → R,  ϕ(r) = arccos r + r 1 − r 2 , satisVes ϕ  (0) = ϕ  (0) = 0 but ϕ  (0) = 0. This implies that ∂ E isop is of class C 2 but not of class C 3 . In fact, we have −2r 2 , ϕ  (r) = √ 1 − r2

ϕ  (r) = −2r

2 − r2 , (1 − r 2 )3/2

and thus ϕ  (0) = −4 = 0. 2) The set E isop is convex because the function ϕ satisVes ϕ  ≤ 0 on [0, 1] and ϕ  (0) = 0. 3) The set E isop is axially symmetric: (z, t) ∈ E isop

⇒

(ζ, t) ∈ E isop

In fact, the proVle function depends on |z|.

for all |ζ | = |z|.

105 Isoperimetric problem and minimal surfaces

Pansu’s conjecture is known to hold assuming some regularity, symmetry, or structure for the isoperimetric set. In the next sections, we describe the following recent results: 1) If E ⊂ H1 is isoperimetric and ∂ E is of class C 2 then E = E isop , up to dilation and left translation. This result is not known when n ≥ 2. 2) If E ⊂ H1 is isoperimetric and convex then E = E isop , up to dilation and left translation. This result is not known when n ≥ 2. 3) Let n ≥ 1. If E ⊂ Hn is isoperimetric and axially symmetric then E = E isop , up to a vertical translation and a dilation. 4) Let n ≥ 1. If E ⊂ Hn is contained in a vertical cylinder and has a circular horizontal section, then E = E isop , up to dilation and left translation. In general, Pansu’s conjecture is still open. 4.2 Isoperimetric sets of class C 2 In this section, we show that isoperimetric sets in H1 of class C 2 are of the form (4.9). This result is due to [69] (Theorems 6.10 and 7.2) and relies upon two facts: the structure of the characteristic set of surfaces of class C 2 ; the geometric interpretation of the equation for surfaces with constant H -curvature. Both results are limited to H1 . Theorem 4.4 (Ritor´e-Rosales). Let E ⊂ H1 be a bounded isoperimetric set with boundary ∂ E of class C 2 . Then we have E = E isop , up to dilation and left translation. Proof. Let D ⊂ C be an open set and let f ∈ C 2 (D) be a function such that   gr( f ) = (z, f (z)) ∈ H1 : z ∈ D ⊂ ∂ E.   We denote by ( f ) = z ∈ D : ∇ f (z) + 2z ⊥ = 0 the characteristic set of f . It may be ( f ) = ∅. We always have |( f )| = 0. For ϕ ∈ Cc∞ (D \ ( f )) and  ∈ R small, consider the set E  ⊂ H1 that is obtained from E perturbing the piece of boundary of E given by the graph of f , through the function f + ϕ. Then, for small  we have P(E  ; H1 ) p() P(E; H1 ) = I (E) ≤ I (E ) = = = ψ(), (4.10)  |E|3/4 |E  |3/4 v()3/4 where p() = P(E  ; H1 ) and p() = |E  |. Using the area formula for H -perimeter (3.5) we Vnd   ∇ f + 2z ⊥ , ∇ϕ   dz, v (0) = − ϕ(z) dz. p (0) = |∇ f + 2z ⊥ | D D

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Here, we are assuming that the set E lies above the graph of f . Moreover, we have ψ  = p v −3/4 − 34 pv −7/4 v  . From (4.10) we deduce that ψ  (0) = 0 and thus   ∇ f + 2z ⊥ , ∇ϕ 3 P(E; H1 ) 1 ϕ dz dz + 0= |E|3/4 D |∇ f + 2z ⊥ | 4 |E|7/4 D   ∇ f + 2z ⊥

1 3 P(E; H1 ) = − 3/4 ϕ div ϕ dz. dz + |E| |∇ f + 2z ⊥ | 4 |E|7/4 D D Since ϕ ∈ Cc∞ (D \ ( f )) is arbitrary, we deduce that the function f satisVes the partial differential equation ∇ f (z) + 2z ⊥ 3 P(E; H1 ) div = =: H, |∇ f (z) + 2z ⊥ | 4 |E|

z ∈ D \ ( f ). (4.11)

We conclude that for any z ∈ D\( f ) there exists an arc of circle κz with curvature H passing through z and such that γz = Lift(κz ) is contained in gr( f ) ⊂ ∂ E. See Remark 3.13. Let (∂ E) be the characteristic set of ∂ E. The above argument shows that for any p ∈ ∂ E \ (∂ E) there exists a geodesic γ p contained in ∂ E \ (∂ E) and passing through p. There exists a maximal interval (a, b) such that we have γ p : (a, b) → ∂ E \ (∂ E). Since E is bounded, γ p can be extended to a and b with γ (a), γ (b) ∈ (∂ E). In a neighborhood of the point (z 0 , t0 ) = γ (a) ∈ ( f ), the surface ∂ E is a graph of the form t = f (z) for some f ∈ C 2 (D) and D ⊂ C open set with z 0 ∈ D. This is because the tangent space to ∂ E at this point coincides with the horizontal plane. Let (D, f ) be the maximal pair such that gr( f ) ⊂ ∂ E with D open set containing z 0 and f ∈ C 2 (D). By Theorem 3.15, there are two cases: i) z 0 is an isolated point of ( f ); ii) Near z 0 , ( f ) is a C 1 curve κz0 passing through z 0 . In the case ii), let κz0 be the maximal C 1 curve contained in ( f ) and passing through z 0 . The curve κz0 cannot reach the boundary ∂ D because this would contradict the maximality of D. The curve κz0 cannot have limit points inside D that are singular, because of Theorem 3.15. Then κz0 must be a simple closed curve inside D. But this is not possible because the horizontal lift of κz0 grows in the t coordinate by an amount that equals 4 times the area of the region enclosed by the simple closed curve. So we are left with the case ( f ) = {z 0 } for some z 0 ∈ D. Through any point z ∈ D \ {z 0 } passes a circle with curvature H starting from z 0 .

107 Isoperimetric problem and minimal surfaces

Now the boundary of E is determined in a neighborhood of (z 0 , f (z 0 )) ∈ ∂ E. The regularity of ∂ E forces D to be a circle centered at z 0 and E to be a left translation and dilatation of E isop . 4.3 Convex isoperimetric sets We say that a set E ⊂ H1 is convex if it is convex for the standard linear structure of H1 = R3 . Left translations and dilations preserve convexity. In [53], Pansu’s conjecture is proved assuming the convexity of isoperimetric sets. Recall the E isop ⊂ H1 is the set in (4.9). Theorem 4.5 (Monti-Rickly). Let E ⊂ H1 be a convex (open) isoperimetric set. Then, up to a left translation and a dilation we have E = E isop . Using the concentration argument of Theorem 4.1, it is possible to prove the existence of isoperimetric sets within the class of convex sets. However, it is not clear how to compute the Vrst variation remaining inside this class of sets. Theorem 4.5 is not known when n ≥ 2. It would be also interesting to prove the theorem assuming for isoperimetric sets only H -convexity (convexity along horizontal lines, see [5]) rather than standard convexity. Here, we describe the technical steps of the proof of Theorem 4.5. For details, we refer the reader to [53]. Let E ⊂ H1 be a convex isoperimetric set. Then we have   (4.12) E = (z, t) ∈ H1 : z ∈ D, f (z) < t < g(z), , where D ⊂ C = R2 is a bounded convex open set in the plane, and −g, f : D → R are convex functions. In particular, f and g are locally Lipschitz continuous and their Vrst derivatives are locally of bounded variation. The function f satisVes the following partial differential equation   ∇ f + 2z ⊥ 3P(E; H1 ) = = H in D. (4.13) div |∇ f + 2z ⊥ | 4|E| Equation (4.13) can be deduced in the same way as in (4.11), with the difference that the equation is now veriVed only in the weak sense. As a matter of fact, the vector Veld N f (z) =

∇ f (z) + 2z ⊥ |∇ f (z) + 2z ⊥ |

z ∈ D,

is only in L ∞ (D). However, we have ∇ f (z) + 2z ⊥ ∈ BVloc (D).

108 Roberto Monti

The goal is to prove that integral curves of N ⊥f are circles with curvature H . The vector N f will be the “normal vector” to the curve. The Vrst step of the proof of Theorem 4.5 is an improved regularity for solutions of (4.13): the candidate “normal vector” satisVes N f ∈ 1,1 Wloc (D; R2 ), see [53]. The second step of the proof consists in the analysis of the Wow of the vector Veld v(z) = 2z − ∇ f ⊥ (z). This vector Veld is orthogonal to N f . Since f is convex, we have v ∈ BVloc (int(D); R2 ). Moreover, the distributional divergence of v is in L ∞ , in fact div v = 4 in int(D). Thus, by Ambrosio’s theory on the Cauchy Problem for vector Velds of bounded variation [2], for any compact set K ⊂ D there exist r > 0 and a (unique regular) Lagrangian Wow  : K × [−r, r] → D. In particular, for any z ∈ K , the curve γz (s) = (z, s) is an integral curve of v passing through z at time s = 0. 1,1 (D; R2 ) to The third step of the proof uses the fact that v/|v| is in Wloc show that (a suitable reparameterization of) the integral curve γz is twice differentiable in a weak sense. With this regularity, the distributional equation (4.13) can be given a formal meaning along the integral curve γz : it says that the curvature of γz is the constant H . Theorem 4.6. Let E ⊂ H1 be a convex isoperimetric set with curvature H > 0 (the constant in (4.13)) and let  : K × [−r, r] → D be the Gow introduced above. Then for a.e. z ∈ K the curve s → (z, s) is an arc of circle with radius 1/H . The shape of a convex isoperimetric set E can now be reconstructed starting from the structure of the characteristic set of ∂ E. A point (z, t) ∈ ∂ E is characteristic if the horizontal plane at (z, t) is a supporting plane for E at (z, t). For convex sets, the characteristic set is the disjoint union of at most four compact disjoint horizontal segments, possibly points, see [53]. This property and Theorem 4.6 yield Theorem 4.5 as explained in the Vnal part of the proof of Theorem 4.4. 4.4 Axially symmetric solutions We denote by S the family of all measurable subsets E ⊂ Hn with 0 < |E| < ∞ that are axially symmetric: (z, t) ∈ E

⇒

(ζ, t) ∈ E

for all |ζ | = |z|.

The isoperimetric problem in the family S consists in proving existence and classifying all minimizers of the inVmum problem   S (4.14) Cisop = inf I (E) : E ∈ S .

109 Isoperimetric problem and minimal surfaces

A set E ∈ S for which the inVmum in (4.14) is attained is called an S axially symmetric isoperimetric set. Clearly, we have Cisop ≥ Cisop . Even S

though we believe that Cisop = Cisop , we are not able to prove this. In the axially symmetric setting, Pansu’s conjecture amounts to show that the solution to Problem (4.14) is the set

  E isop = (z, t) ∈ Hn : |t| < arccos |z| + |z| 1 − |z|2 , |z| < 1 . (4.15) for any dimension n ≥ 1. This result is proved in [48] and, in this section, we present the scheme of the proof. S

Theorem 4.7 (Monti). The inFmum Cisop > 0 is attained and any axially symmetric isoperimetric set coincides with the set E isop in (4.15), up to a dilation, a vertical translation, and a Lebesgue negligible set. By a rearrangement argument, Theorem 4.7 can be reduced to a one dimensional problem. The Vrst step is the reduction to an isoperimetric problem in the half plane R2+ = R+ × R. Using spherical coordinates in Cn , a measurable axially symmetric set E ⊂ Hn is generated by a measurable set F ⊂ R2+ (and viceversa), and we have the following formula P(E; Hn ) = ω2n−1 Q(F; R2+ ),

(4.16)

where Q(·; R2+ ) is a weighted perimeter functional in the half-plane    2n−1    Q(F; R2+ ) = sup ∂r r ψ1 + ∂t 2r 2n ψ2 drdt : ψ F (4.17)  1 2 2 ∈ Cc (R+ ; R ), ψ∞ ≤ 1 . Above, ω2n−1 = H 2n−1 (S2n−1 ) is the standard surface measure of the (2n −1)-dimensional unit sphere. For any axially symmetric set E ⊂ Hn , the volume transforms according to the following rule  (4.18) |E| = ω2n−1 r 2n−1 drdt = ω2n−1 V (F), F

where V (·) is a volume functional in the half-plane. From (4.17) and (4.18), the axially symmetric isoperimetric problem (4.14) transforms into the weighted isoperimetric problem in the half plane ( ) Q(F; R2+ ) S 1/Q : F ⊂ R2+ such that 0 < V (F) < ∞ . Cisop = ω2n−1 inf Q−1 Q V (F) (4.19)

110 Roberto Monti

The observation made in [48] is that the isoperimetric quotient for sets F ⊂ R2+ is improved by a certain rearrangement of F in the variable r for Vxed t that is tailored to the perimeter Q(·;R2+ ). We measure the t-sections of F, the sets Ft = r > 0 : (r, t) ∈ F , using the line density τ (r) = 2r 2n . The function τ is the weight appearing in the deVnition of the functional Q(·; R2+ ) in (4.17). We let  !(r) =

r

τ (s) ds =

0

2 r 2n+1 , 2n + 1

(4.20)

and we say that a measurable set F ⊂ R2+ is τ -rearrangeable if the function f : R → [0, +∞]  τ (r) dr

f (t) =

(4.21)

Ft

is in L 1loc (R). In this case, we call the set   F " = (r, t) ∈ R2+ : !(r) < f (t)

(4.22)

the τ -rearrangement of F. The t-sections of F " are intervals (0,!−1 ( f (t))) with the same τ -measure as the t-sections Ft . The following intermediate result is proved in [48]. Theorem 4.8. Let F ⊂ R2+ be a τ -rearrangeable set. Then: i) We have Q(F " ; R2+ ) ≤ Q(F; R2+ ), and in case of equality there holds F = F " , up to a negligible set. ii) We have V (F " ) ≥ V (F). Using Theorem 4.8, it is easy to Vnd a compact minimizing sequence, thus getting the existence of axially symmetric isoperimetric sets. Moreover, a set F minimizing (4.19) satisVes: i) F = F " , up to a negligible set; ii) the sections Fr = {t ∈ R : (r, t) ∈ F} are equivalent to intervals, for L 1 -a.e. r ∈ R+ . Now the boundary of ∂ F inside R2+ is a Lipschitz curve that can be computed by a standard variational argument. This curve is the proVle of the isoperimetric set conjectured by Pansu and, as a matter of fact, it does not depend on the dimension n.

111 Isoperimetric problem and minimal surfaces

4.5 Calibration argument In [67], Ritor´e proved Pansu’s conjecture within a special class of sets by a calibration argument. The sets have one circular horizontal section and are contained in a vertical cylinder, see also [19]. The argument works in any dimension. We let B = {(z, 0) ∈ Hn : |z| < 1} and C = {(z, t) ∈ Hn : |z| < 1, t ∈ R}. We identify B = {|z| < 1} ⊂ Cn . Theorem 4.9 (Ritor´e). Let E ⊂ Hn , n ≥ 1, be a bounded open set with Fnite H -perimeter such that: i) B ⊂ E ⊂ C; ii) |E| = |E isop |, where E isop is the set in (4.15). Then, we have P(E isop ; Hn ) ≤ P(E; Hn ). Proof. Let ϕ : B¯ → R be the proVle function of E isop ,  ϕ(z) = arccos |z| + |z| 1 − |z|2 , |z| ≤ 1. The function f : C¯ → R, f (z, t) = |t| − ϕ(z), is a deVning function for ∂ E isop . Let us deVne the vector Veld ψ : C¯ \ Z → R2n ψ(z, t) =

∇H f (z, t) , |∇H f (z, t)|

0 < |z| < 1,

t  = 0.

The vector Veld ψ is not deVned when z = 0 or t = 0; it can be extended to |z| = 1; it jumps at t = 0. In the set {0 < |z| < 1, t  = 0}, ψ is of class C ∞ and there is a constant H = 0 such that divH ψ(z, t) = H,

0 < |z| < 1,

t  = 0.

(4.23)

We consider the following sets: E + = E ∩ {t > 0},

+ E isop = E isop ∩ {t > 0}

E − = E ∩ {t < 0},

− E isop = E isop ∩ {t < 0}.

+ + ⊂ C and moreover the boundary of E + E isop By i), we have E + E isop + does not intersect the base B of the cylinder. Let F + = E isop \ E + and + + G + = E + \ E isop . Then we have F + , G + ⊂ B × R+ and E + E isop = + + G F + + F ∪ G . Moreover, denoting by N H and N H the horizontal outer normals to ∂ F + and ∂G + , respectively: +

E

+

a) N HF = N H isop a.e. on ∂ F + ∩∂ E isop and N HF = −N HE a.e. on ∂ F + ∩∂ E;

112 Roberto Monti

+

+

E

b) N HG = −N H isop a.e. on ∂G + ∩ ∂ E isop and N HG = N HE a.e. on ∂G + ∩ ∂ E. Integrating (4.23) on F + we Vnd   + + divH ψ(z, t) dzdt = N HF , ψdμ F + H |F | = + + F ∂ F E isop = N H , ψdμ Eisop − N HE , ψdμ E ∂ F+

(4.24)

∂ F+ +

≥ P(E isop ; ∂ F + ) − P(E; ∂ F ), E

because N H isop , ψ = 1 on ∂ F + ∩∂ E isop and N HE , ψ ≤ 1 on ∂ F + ∩∂ E. In the same way, we Vnd the inequality   + H |G + | = divH ψ(z, t) dzdt = N HG , ψdμG + + + G ∂G   E isop (4.25) =− N H , ψdμ Eisop + N HE , ψdμ E ∂G +

+

∂G + +

≤ −P(E isop ; ∂G ) + P(E; ∂G ). From (4.24) and (4.25), we obtain

H (|F + | − |G + |) ≥ P(E isop ; ∂ F + ) − P(E; ∂ F + ) + P(E isop ; ∂G + ) − P(E; ∂G + ) = P(E isop ; {t > 0}) − P(E; {t > 0}).

(4.26)

− − . Then we have F − , G − ⊂ \ E − and G − = E − \ E isop Let F − = E isop − B × R− and E − E isop = F − ∪ G − . Computations analogous to the ones above show that

H (|F − | − |G − |) ≥ P(E isop ; {t < 0}) − P(E; {t < 0}).

(4.27)

Since |E| = |E isop | we have |F + | + |F − | = |G + | + |G − |. Adding (4.26) and (4.27), we obtain 0 = H (|F + | + |F − | − |G + | − |G − |) ≥ P(E isop ; {t = 0}) − P(E; {t  = 0}) = P(E isop ; Hn ) − P(E; Hn ). This concludes the proof. Remark 4.10. In [67], Ritor´e also discusses the equality case. Namely, in the setting of Theorem 4.9 and assuming that ∂ E \ Z is a C H1 -regular surface, he shows that the equality P(E; Hn ) = P(E isop ; Hn ) implies E = E isop .

113 Isoperimetric problem and minimal surfaces

5 Regularity problem for H -perimeter minimizing sets The regularity of H -perimeter minimizing boundaries is a challenging open problem. We list the main steps and the main technical difVculties. 1) Lipschitz approximation. The Vrst step in the regularity theory of perimeter minimizing sets in Rn is a good approximation of minimizers. In De Giorgi’s original approach, the approximation is made by convolution and the estimates are based on the monotonicity formula. In the Heisenberg group, the validity of a monotonicity formula is not clear, see [21]. A more Wexible approach is the approximation of minimizing boundaries by Lipschitz graphs. This scheme works also in the Heisenberg group. An H -minimizing boundary is approximated in measure by an intrinsic Lipschitz graph. The estimate involves the notion of horizontal excess, see Theorem 5.9 and [50]. 2) Harmonic approximation. The minimal set can be blown-up at a point of the reduced boundary by a quantity depending on excess. It can be shown that the corresponding approximating intrinsic Lipschitz functions converge to a limit function. This holds when n ≥ 2 thanks to a Poincar´e inequality valid on vertical hyperplanes, see [17]. We do not present the details, here. It is an open problem to prove that this limit function is harmonic for the natural (linear) sub-Laplacian of the vertical hyperplane. 3) Decay estimate for excess. Known estimates for sub-elliptic harmonic functions should give the decay estimate for excess Exc(E, Bαr ) ≤ Cα 2 Exc(E, Br ),

r > 0,

for some 0 < α < 1 and C > 0. By standard facts, this implies the H¨older continuity of the horizontal normal on the reduced boundary. In turn, the continuity of the normal implies that the reduced boundary is a C H1 -regular surface in the sense of DeVnition 2.16, see [56], and thus it is locally the intrinsic graph of a continuous function ϕ having H¨older continuous distributional intrinsic gradient ∇ ϕ ϕ, see DeVnition 3.4. 4) Schauder-type regularity. The function ϕ is a local minimizer of the area functional (see (3.14))   1 + |∇ ϕ ϕ|2 dw. A (ϕ) = D

It is an open problem to deduce further regularity for ϕ, beyond the H¨older continuity of the distributional gradient ∇ ϕ . It is not even clear how to prove that ϕ solves the minimal surface equation (3.25). This is the state of the art on the regularity of H -perimeter minimizing boundaries. In Section 5.3, we present the Lipschitz approximation of

114 Roberto Monti

H -perimeter minimizing sets, Theorem 5.9, and also the so-called height estimate, giving a certain Watness of the boundary in the regime of small excess, see Theorem 5.10. The proofs are rather technical and are omitted. In Section 5.2, we also study some examples of nonsmooth minimizers in H1 , including sets with constant horizontal normal. No similar examples of nonsmooth minimizers are known in Hn with n ≥ 2. 5.1 Existence and density estimates We start from the deVnition of a local minimizer of H -perimeter. DeHnition 5.1. A set E ⊂ Hn with locally Vnite H -perimeter in an open set A ⊂ Hn is H -perimeter minimizing in A if for all p ∈ Hn and r > 0 and for any F ⊂ Hn such that EF  Br ( p)  A we have P(E; Br ( p)) ≤ P(F; Br ( p)).

(5.1)

The existence of local minimizers with some boundary condition easily follows by a compactness argument. Let A ⊂ Hn be a bounded open set and let B ⊂ Hn be a set such that P(B; Hn ) < ∞. DeVne the family of sets:   F(A, B) = F ⊂ Hn : F has Vnite H -perimeter in Hn and FB ⊂ A¯ . Clearly, F(A, B) = ∅ because B ∈ F(A, B). The set B determines a natural boundary condition. Proposition 5.2. Let A and B be as above. Then there exists a set E ∈ F(A, B) such that P(E; Hn ) ≤ P(F; Hn )

for all F ∈ F(A, B).

Proof. DeVne the inVmum   m = inf P(F; Hn ) : F ∈ F(A, B) ≥ 0, and let (E j ) j∈N be a minimizing sequence of sets E j ∈ F(A, B): lim P(E j ; Hn ) = m.

j→∞

Let  ⊂ Hn be a bounded open set such that A¯ ⊂  and supporting the compact embedding BVH ()  L 1 (). The C 2 regularity of the boundary ∂ is a sufVcient condition for compactness (see [33] and [52]).

115 Isoperimetric problem and minimal surfaces

Then we have: i) P(E j ; Hn ) ≤ m + 1 for all j ∈ N large enough; ii) |E j ∩ | ≤ || < ∞ for all j ∈ N. By compactness, there exists a subsequence, still denoted by (E j ) j∈N , and a measurable set E ⊂ Hn such that χ E j → χ E in L 1 (). Since ¯ we can also assume that χ E = χ B in Hn \ A, ¯ that is χ E j = χ B in Hn \ A, 1 n E ∈ F(A, B). In particular, we have χ E j → χ E in L (H ). By the lower semicontinuity of perimeter for the L 1 convergence of sets, we obtain P(E; Hn ) ≤ lim inf P(E j ; Hn ) = m, j→∞

If now F is a set such that EF  Br ( p)  A, then F ∈ F(A, B) and P(E; Hn \ B¯ r ( p)) = P(F; Hn \ B¯ r ( p)). Therefore, we have P(E; Br ( p)) = P(E; Hn ) − P(E; Hn \ B¯ r ( p)) ≤ P(F; Hn ) − P(F; Hn \ B¯ r ( p)) = P(F; Br ( p)). As for the standard perimeter, sets that are H -perimeter minimizing admit lower and upper density estimates with geometric constants. Lemma 5.3. If E ⊂ Hn is an H -perimeter minimizing set in a ball B for some  > 0, then we have P(E; B ) ≤ c1  Q−1 ,

(5.2)

where c1 = P(B1 ; Hn ). Proof. Let 0 < s < r < . Since the sets E and E \ Bs agree inside Br \ B¯ s , we have P(E; Br \ B¯ s ) = P(E \ Bs ; Br \ B¯ s ) = P(E \ Bs ; Br ) − P(E \ Bs ; B¯ s ). On the other hand, using P(E \ Bs ; Bs ) = 0 and (2.13) we obtain P(E \ Bs ; B¯ s ) = P(E \ Bs ; ∂ Bs ) = cn SQ−1 (∂ ∗ (E \ Bs ) ∩ ∂ Bs ) ≤ cn SQ−1 (∂ Bs ) = P(Bs ; Hn ) = c1 s Q−1 . The formula P(Bs ; Hn ) = s Q−1 P(B1 ; Hn ) follows by an elementary homogeneity argument. Then we obtain the inequality P(E \ Bs ; Br ) ≤

116 Roberto Monti

P(E; Br \ B¯ s ) + c1 s Q−1 . Since E is H -perimeter minimizing in B , by (5.1) we get P(E; Br ) ≤ P(E \ Bs ; Br ) ≤ P(E; Br \ B¯ s ) + c1 s Q−1 . Letting s ↑ r and using P(E; Br ) < ∞, we obtain P(E; Br ) ≤ c1r Q−1 . Letting r ↑ , we obtain (5.2). The density estimates from below are proved in [71], Proposition 2.14 (see also Theorem 2.4 therein). Lemma 5.4. There exist constants c2 , c3 > 0 depending on n ≥ 1 such that for any set E ⊂ Hn that is H -perimeter minimizing in B2 ,  > 0, we have, for all p ∈ ∂ E ∩ B and for all 0 < r < ,   (5.3) min |E ∩ Br ( p)|, |Br ( p) \ E| ≥ c2r Q , and

P(E; Br ( p)) ≥ c3r Q−1 .

(5.4)

For any set, the reduced boundary is a subset of the measure theoretic boundary, ∂ ∗ E ⊂ ∂ E, and moreover μ E (∂ E \ ∂ ∗ E) = 0, see Proposition 2.8. For local minimizers the difference ∂ E \ ∂ ∗ E is also small in terms of Hausdorff measures. Lemma 5.5. For any set E ⊂ Hn that is H -perimeter minimizing in Hn , we have   (5.5) SQ−1 ∂ E \ ∂ ∗ E = 0. Proof. Let K = ∂ E \ ∂ ∗ E, let A be an open set containing K , and Vx δ > 0. For any p ∈ K there is an 0 < r p < δ/10 such that B5r p ( p) ⊂ A. Then {Br p ( p) : p ∈ K } is a covering of K and by the 5-covering lemma, there exists a sequence pi ∈ K , i ∈ N, such that the balls Bi = Bri ( pi ), with ri = r pi , are pairwise disjoint and K ⊂

B5ri ( pi ). i∈N

It follows that SQ−1,δ (K ∩ A) ≤ ≤



diam(B5ri ( pi )) Q−1 = 10 Q−1

i∈N 10 Q−1 c3−1

 i∈N

P(E; Bri ( pi )) ≤



riQ−1

i∈N 10 Q−1 c3−1 P(E;

A).

Since δ > 0 is arbitrary, we deduce that SQ−1 (K ) ≤ 10 Q−1 c3−1 P(E; A). As A is arbitrary and, by (2.13), P(E; K ) = 0, we conclude that SQ−1 (K ) = 0.

117 Isoperimetric problem and minimal surfaces

5.2 Examples of nonsmooth H -minimal surfaces The existence of nonsmooth H -minimal surfaces in H1 was already observed in [61]. Then this phenomenon was noticed by several authors, see [14, 55, 66, 71]. In the next examples, we prove perimeter minimality of certain H -minimal surfaces by a calibration argument, see [8, 55]. 5.2.1 A Lipschitz H -minimal surface In this example, we study a local minimizer of H -perimeter with boundary ∂ E that is only Lipschitzregular. The surface ∂ E is, however, C H1 -regular: whereas the standard normal jumps, the horizontal normal is continuous.   In the open half-space A = (z, t) ∈ H1 : y = Im(z) > 0 , consider the set   E = (z, t) ∈ A : x = Re(z) < 0 and t < 0 .

The set E has locally Vnite H -perimeter in A and its boundary S = ∂ E ∩ A is a Lipschitz surface consisting of two pieces of plane meeting at the singular line L = {(z, t) ∈ A : x = 0 and t = 0 . The horizontal inner normal ν E : S → R2 is the restriction to S of the mapping ϕ : A → R2 ⎧ ⎨ (−y, x) if x ≤ 0, 2 2 ϕ(z, t) = ⎩ x +y (−1, 0) if x ≥ 0. The function ϕ is continuous in A and thus S is an H -regular surface. In fact, ϕ is locally Lipschitz continuous in A. We claim that E is a local minimizer of H -perimeter in A. Namely, we prove that for any bounded open set  ⊂⊂ A and for any F ⊂ A such that EF   we have P(E; ) ≤ P(F; ).

(5.6)

The proof is a calibration argument and the calibration is provided by the vector Veld V in A deVned by V (z, t) = ϕ1 (z, t)X + ϕ2 (z, t)Y, where ϕ = (ϕ1 , ϕ2 ). Then, at points (z, t) ∈ A where x ≤ 0 we have −y



x div V = divH ϕ = X  +Y  x 2 + y2 x 2 + y2

∂ −y

∂ x   = + = 0. ∂x ∂y x 2 + y2 x 2 + y2 Trivially, we have div V = 0 where x ≥ 0.

118 Roberto Monti

Without loss of generality, we assume that F is closed, that ∂ F ∩ A is a smooth (say, Lipschitz) surface, and that F \ E = ∅ in such a way that EF = E \ F = E ∩ F  , where F  = Hn \ F.  Let N E , N F , and N E∩F denote the Euclidean outer unit normals to the boundary of ∂ E, ∂ F, and ∂(E ∩ F  ), respectively. By the divergence theorem, we have 

 0=

E∩F 

div V dzdt =



∂(E∩F  )

V, N E∩F dH

2

 =

∂ E∩F 



−

V, N E dH

∂ F∩E

2

(5.7)

V, N F dH 2 .

On ∂ E, we have   X, N E , Y, N E  , ϕ= X, N E 2 + Y, N E 2 and thus V, N E  = ϕ1 X, N E  + ϕ2 Y, N E  =

 X, N E 2 + Y, N E 2 .

By the area formula (3.2), it follows that 

 ∂ E∩F 

V, N E dH 2 =

∂ E∩F 

 X, N E 2 + Y, N E 2 dH

2

= P(E; F  ).

On the other hand, on ∂ F we have |ϕ| = 1 and by the Cauchy-Schwarz inequality we obtain V, N F  = ϕ1 X, N F  + ϕ2 Y, N F  ≤ So we deduce that   F 2 V, N dH ≤ ∂ F∩E

∂ F∩E

 X, N F 2 + Y, N F 2 .

 X, N F 2 + Y, N F 2 dH 2 = P(F; E).

So (5.7) implies P(E; F  ) ≤ P(F; E), and this is equivalent to (5.6).

119 Isoperimetric problem and minimal surfaces

5.2.2 An H -minimal intrinsic graph with discontinuous normal In this example, we study an H -minimal intrinsic Lipschitz graph with discontinuous horizontal normal. This surface is a t-graph with standard C 1,1 regularity. √ Let ϕ : R2 → R be the function ϕ(y, t) = sgn(t) |t|. The intrinsic epigraph of ϕ in the sense of DeVnition 3.3 is the set

  E = (s, y, t + 2ys) ∈ H1 : (y, t) ∈ R2 , s > ϕ(y, t) . The boundary of E is the intrinsic graph of ϕ:   ∂ E = (ϕ(y, t), y, t + 2yϕ(y, t)) ∈ H1 : (y, t) ∈ R2 . The intrinsic gradient of ϕ in the sense of DeVnition 3.4 reduces to the Burgers’ component ∇ ϕ ϕ = Bϕ = ϕ y − 4ϕϕt = −2sgn(t),

t  = 0.

Then ∇ ϕ ϕ ∈ L ∞ (R2 ) and gr(ϕ) is an intrinsic Lipschitz graph, see Theorem 3.9. Moreover, by formula (3.13) the horizontal normal to ∂ E is  (1, −∇ ϕ ϕ) 1  = √ 1, 2sgn(t) . νE =  ϕ 2 5 1 + |∇ ϕ| The normal can be extended in a constant way to H1 \ {x  = 0}, when x > 0 and x < 0, separately. √ Letting x = sgn(t) |t|, we realize that ∂ E is the t-graph of the function f : R2 → R, f (x, y) = x|x| + 2x y: ∂E =



  x, y, f (x, y) ∈ H1 : (x, y) ∈ R2 .

Clearly, we have f ∈ C 1,1 (R2 ). We claim that E is a local minimizer for H -perimeter in H1 . Namely, we prove that for any bounded open set A ⊂ H1 and for any measurable set F ⊂ H1 with locally Vnite H -perimeter and such that EF  A there holds P(E; A) ≤ P(F; A). (5.8) Without loss of generality, we assume that ∂ F ∩ A is a smooth surface. Let G = EF and consider the subsets of G: G − = (EF) ∩ {x < 0} and

G + = (EF) ∩ {x > 0}.

120 Roberto Monti

Let N E , N F , N G be the Euclidean outer normals to ∂ E, ∂ F, and ∂G, respectively. To Vx ideas, we assume that F \ E = ∅, so that we have ¯ a.e. on ∂ E ∩ G, ¯ N G = −N F a.e. on ∂ F ∩ G.

NG = NE

√ DeVne the horizontal vector Veld V in H1 by V = 5(ν E1 X + ν E2 Y ), where ν E = (ν E1 , ν E2 ) is the extended horizontal normal. Namely, we let  V =

X − 2Y x < 0 X + 2Y x > 0.

The vector Veld V is not deVned on the plane x = 0. When x  = 0 we have divV = divH ν E = 0. By the divergence theorem applied to G − and G + , we obtain    divV dzdt = divV dzdt + divV dzdt 0= G− G+  G V, NG − dH 2 + V, NG + dH 2 . = ∂G −

∂G +

We denote by V − and V + the traces of V onto {x = 0}, from the left and from the right. The integral on ∂G − is    V, NG − dH 2 = V, N G dH 2 + V − , N G dH 2 − ∂G ∂G∩{x 0, with 0 ∈ ∂ E. Assume that ν E ( p) = (1, 0) ∈ S1 for μ E -a.e. p ∈ Q 4r . Then there exists a function g : Dr → (−r/4, r/4) such that: i) We have, up to a negligible set,   E ∩ Q r = (x, y, t) ∈ Q r : x > g(y, t) . ii) g(0) = 0 and for all (y, t), (y  , t  ) ∈ Dr |g(y, t) − g(y  , t  )| ≤ |y − y  | +

1 |t − t  |. 2r

iii) The graph of g consists of integral lines of the vector Feld Y .

(5.11)

122 Roberto Monti

Proof. For the sake of simplicity, we assume that E is open. For any α, β ∈ R with α ≥ 0, let Z = α X + βY . Then, for any ϕ ∈ Cc1 (Q 4r ) with ϕ ≥ 0, by the Gauss-Green formula (2.6) we have   Z ϕ dzdt = −α ϕ dμ E ≤ 0, E

Q 4r

that is Z χ E ≥ 0 in the sense of distribution. It follows that p ∈ E ∩ Q 4r

⇒

exp(s Z )( p) ∈ E,

(5.12)

for all s > 0 such that exp(s Z )( p) ∈ Q 4r . For any point q ∈ E ∩ Q 2r consider the set E q = q −1 · E. The set E q has constant measure theoretic normal (1, 0) ∈ S1 in Q 2r . We can apply (5.12) to the set E q starting Vrst from the point 0 ∈ E q and then from a generic point p = (0, y, 0) ∈ E q with |y| < 2r. We deduce that   (x, y, t) ∈ Q 2r : x > 0, |t| < 4r x ⊂ E q . In other words, we have   q · (x, y, t) ∈ Q 2r : x > 0, |t| < 4r x ⊂ E. (5.13) From (5.13), it follows that E ∩ Q 2r ∩ {y = 0} is a planar set with the cone property, the cones having all axis parallel to the x-axis and aperture 4r. We deduce that there exists a Lipschitz function h : (−r 2 , r 2 ) → R such that:     (a) (x, t) ∈ R2 : (x, 0, t) ∈ E = (x, t) ∈ D2r : x > h(t) ; 1 (b) |h(t) − h(t  )| ≤ |t − t  | for all t, t  ∈ (−r 2 , r 2 ). 4r q ∈ E ∩ Q 2r

⇒

Since 0 ∈ ∂ E, we infer that h(0) = 0. From (5.13), we also deduce that ∂ E consists of integral lines of Y in Q 2r . Then we have   ∂ E ∩ Q 2r = (h(τ ), σ, τ − 2σ h(τ )) ∈ H1 : (σ, τ ) ∈ D2r .

(5.14)

For any (y, t) ∈ Dr , the system of equations σ = y,

τ − 2σ h(τ ) = t

has a unique solution (σ, τ ) ∈ D2r . This is an easy consequence of the Banach Vxed point theorem. We claim that the solution τ = τ (y, t) of

123 Isoperimetric problem and minimal surfaces

the equation τ − 2yh(τ ) = t is Lipschitz continuous. Namely, by (b), we have for (y, t), (y  , t  ) ∈ Dr |τ (y, t) − τ (y  , t  )| = |t − 2yh(τ (y, t)) − t  + 2y  h(τ (y  , t  ))| ≤ |t − t  | + 2|y||h(τ (y, t)) − h(τ (y  , t  ))| + 2|h(τ (y  , t  ))||y − y  | 1 ≤ |t − t  | + |τ (y, t) − τ (y  , t  )| 2 1   + |τ (y , t )||y − y  |, 2r and this implies |τ (y, t) − τ (y  , t  )| ≤ 4r|y − y  | + 2|t − t  |.

(5.15)

The function g = h ◦ τ satisVes i), ii), and iii). In particular, (5.11) follows from (5.15), and |g(y, t)| < r/4 follows from (b). There are H -perimeter minimizing surfaces in H1 with a diffuse Lipschitz regularity. In fact, if g : Dr → (−r/4, r/4) is a function satisfying ii) and iii) of Theorem 5.6, then its x-graph is a Lipschitz surface that has, H 2 -a.e., constant horizontal normal. This vector can be used to show that the x-graph of g is locally minimizing H -perimeter. Remark 5.7. If, in Theorem 5.6, the radius r can be taken arbitrarily large, then from (5.11) we deduce that the function g does not depend on t. Then from statement iii), we deduce that g does not depend on y, either. Thus E is a vertical half-space. This fact is used in Theorem 2.10. When n ≥ 2, the situation is different and easier because if the horizontal normal ν E is constant in a small convex set then, inside this set, E is a vertical hyperplane orthogonal to the normal (see [27]). 5.3 Lipschitz approximation and height estimate The notion of horizontal excess is natural: DeHnition 5.8 (Horizontal excess). Let E ⊂ Hn be a set with locally Vnite H -perimeter. The horizontal excess of E in a ball Br ( p), where p ∈ Hn and r > 0, is  1 |ν E − ν|2 dμ E . Exc(E, Br ( p)) = min Q−1 ν∈R2n r Br ( p) |ν|=1

Intrinsic Lipschitz graphs are introduced in DeVnition 3.3, the notion of L-intrinsic Lipschitz function is introduced in DeVnition 3.8. The following theorem is proved in [50].

124 Roberto Monti

Theorem 5.9 (Monti). Let n ≥ 1 and let L > 0 be a constant that is suitably large when n = 1. There are constants k > 1 and c(L , n) > 0 with the following property. For any set E ⊂ Hn that is H -perimeter minimizing in Bkr with 0 ∈ ∂ E and r > 0, there exist ν ∈ R2n with |ν| = 1 and an L-intrinsic Lipschitz function ϕ : Hν → R such that   SQ−1 (gr(ϕ)∂ E) ∩ Br ≤ c(L , n)(kr) Q−1 Exc(E, Bkr ). (5.16) The following extension of the so-called “height estimate” to H -perimeter minimizing sets will be proved in the forthcoming paper [57]. Let ν = (1, 0 . . . , 0) ∈ R2n and let W = ∂ Hν ⊂ Hn be the vertical hyperplane orthogonal to ν, i.e., W = {x1 = 0}. For any r > 0 we let   Dr = w ∈ W : w∞ < r , and we deVne the truncated cylinder over Dr   Cr = Dr · (−r, r) = w · (sν) ∈ Hn : |s| < r . The ν-directional excess of E inside the cylinder Dr is  1 |ν E − ν|2 dμ E . Exc(E, Cr , ν) = Q−1 r Cr Theorem 5.10 (Monti-Vittone). Let n ≥ 2. There exist constants 0 > 0, c0 > 0, and k > 0 such that if E ⊂ Hn is an H -perimeter minimizing set in Ckr with Exc(E, Ckr , ν) ≤ 0 , then we have

 sup |x1 | = |Re(z 1 )| ∈ R : (z, t) ∈ ∂ E ∩ Cr ≤ c0rExc(E, Ckr

1 , ν) 2(Q−1) .

(5.17)

The proof follows the scheme of [70]. It relies on a nontrivial slicing technique and on a lower dimensional isoperimetric inequality. The estimate (5.17) does not hold when n = 1 because of the examples of Section 5.2.3, for which Exc(E, Br ) = 0 but ∂ E is not Wat.

ACKNOWLEDGEMENTS . We wish to thank Luca Capogna, Valentina Franceschi, Gian Paolo Leonardi, Valentino Magnani, Andrea Malchiodi, Francesco Serra Cassano, Manuel Ritor´e, and Davide Vittone for their comments on a preliminary version of the notes.

125 Isoperimetric problem and minimal surfaces

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Regularity of higher codimension area minimizing integral currents Emanuele Spadaro

Abstract. This lecture notes are an expanded and revised version of the course Regularity of higher codimension area minimizing integral currents that I taught at the ERC-School on Geometric Measure Theory and Real Analysis, held in Pisa, September 30th - October 30th 2013. The lectures aim to explain partially without proofs the main steps of a new proof of the partial regularity of area minimizing integer rectiVable currents in higher codimension, due originally to F. Almgren, which is contained in a series of papers in collaboration with C. De Lellis (University of Z¨urich).

Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 1.1. Integer rectiVable currents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 1.2. Partial regularity in higher codimension . . . . . . . . . . . . . . . . . . . . 135 2. The blowup argument: a glimpse of the proof . . . . . . . . . . . . . . . . 136 2.1. Flat tangent cones do not imply regularity . . . . . . . . . . . . . . . . . . 136 2.2. Non-homogeneous blowup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 2.3. Multiple valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 2.4. The need of centering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 2.5. Excluding an inVnite order of contact . . . . . . . . . . . . . . . . . . . . . . 139 2.6. The persistence of singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 2.7. Sketch of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 3. Q-valued functions and rectiVable currents . . . . . . . . . . . . . . . . . . 141 3.1. Q-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 3.2. Graph of Lipschitz Q-valued functions . . . . . . . . . . . . . . . . . . . . 144 3.3. Approximation of area minimizing currents . . . . . . . . . . . . . . . . 145 4. Selection of contradiction’s sequence. . . . . . . . . . . . . . . . . . . . . . . .147 5. Center manifold’s construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.1. Notation and assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 5.2. Whitney decomposition and interpolating functions . . . . . . . . . 153 5.3. Normal approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.4. Construction criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.5. Splitting before tilting. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .159

132 Emanuele Spadaro

5.6. Intervals of Wattening . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.7. Families of subregions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162 6. Order of contact. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .164 6.1. Frequency function’s estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 6.2. Boundness of the frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 7. Final blowup argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 7.1. Convergence to a Dir-minimizer . . . . . . . . . . . . . . . . . . . . . . . . . . 178 8. Open questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

1 Introduction The subject of this course is the study of the regularity of minimal surfaces, considered in the sense of area minimizing integer rectiFable currents. This is a very classical topic and stems from many diverse questions and applications. Among the most known there is perhaps the so called Plateau problem, consisting in Vnding the submanifolds of least possible volume among all those submanifolds with a Vxed boundary. Plateau problem. Let M be a (m + n)-dimensional Riemannian manifold and  ⊂ M a compact (m − 1)-dimensional oriented submanifold. Find an m-dimensional oriented submanifold  with boundary  such that volm () ≤ volm (  ), for all oriented submanifolds   ⊂ M such that ∂  = . It is a well-known fact that the solution of the Plateau problem does not always exist. For example, consider M = R4 , n = m = 2 and  the smooth Jordan curve parametrized in the following way:    = (ζ 2 , ζ 3 ) : ζ ∈ C, |ζ | = 1 ⊂ C2  R4 , where we use the usual identiVcation between C2 and R4 , and we choose the orientation of  induced by the anti-clockwise orientation of the unit circle |ζ | = 1 in C. It can be shown (and we will come back to this point in the next sections) that there exist no smooth solutions to the Plateau problem for such Vxed boundary, and the (singular) immersed 2-dimensional disk   S = (z, w) : z 3 = w2 , |z| ≤ 1 ⊂ C2  R4 , oriented in such a way that ∂ S = , satisVes H2 (S) < H2 (),

133 Higher codimension integral currents

for all smooth, oriented 2-dimensional submanifolds  ⊂ R4 with ∂ = . Here and in the following we denote by Hk the k-dimensional Hausdorff measure, which for k ∈ N corresponds to the ordinary k-volume on smooth k-dimensional submanifolds. This fact motivates the introduction of weak solutions to the Plateau problem, and the main questions about their existence and regularity. 1.1 Integer rectiHable currents One of the most successful theories of generalized submanifolds is the one by H. Federer and W. Fleming in [19] on integer rectiVable currents (see also [8, 9] for the special case of codimension one generalized submanifolds). From now on, in order to keep the technicalities to a minimum level, we assume that our ambient Riemannian manifold M is Euclidean. DeHnition 1.1 (Integer rectiHable currents). An integer rectiVable current T of dimension m in Rm+n is a triple T = (R, τ, θ) such that: (i) R is a rectiFable set, i.e. R = i∈N Ci with Hm (R0 ) = 0 and Ci ⊂ Mi for every i ∈ N \ {0}, where Mi are m-dimensional oriented C 1 submanifolds of Rm+n ; (ii) τ : R → m is a measurable map, called orientation, taking values in the space of m-vectors such that, for Hm -a.e. x ∈ Ci , τ (x) = v1 ∧ · · · ∧ vm with {v1 , . . . , vm } an oriented orthonormal basis of Tx Mi ; (iii) θ : R → Z is a measurable function, called multiplicity, which is integrable with respect to Hm . An integer rectiVable current T = (R, τ, θ) induces a continuous linear functional (with respect to the natural Fr´echet topology) on smooth, compactly supported m-dimensional differential forms ω, denoted by D m , acting as follows  θ ω, τ  dHm .

T (ω) = R

Remark 1.2. The continuous linear functionals deVned in the Fr´echet space D m are called m-dimensional currents. Remark 1.3. Note that the submanifold Mi in DeVnition 1.1 are only C 1 regular. This restriction is not redundant, but it is connected to several aspects of the theory of rectiVable sets. For an integer rectiVable current T , one can deVne the analog of the boundary and the volume for smooth submanifolds.

134 Emanuele Spadaro

DeHnition 1.4 (Boundary and mass). Let T = (R, τ, θ) be an integer rectiVable current in Rm+n of dimension m. The boundary of T is deVned as the (m − 1)-dimensional current acting as follows ∂ T (ω) := T (dω) ∀ ω ∈ D m−1 . The mass of T is deVned as the quantity  M(T ) := |θ| dHm . R

Note that, in the case T = (, τ , 1) is the current induced by an oriented submanifold  with boundary ∂, with τ a continuous orienting vector for  and similarly τ∂ for its boundary, then by Stoke’s Theorem ∂ T = (∂, τ∂ , 1) and M(T ) = volm (). Finally we recall that the space of currents is usually endowed with the weak* topology (often called in this context weak topology). DeHnition 1.5 (Weak topology). We say that a sequence of currents (Tl )l∈N weakly converges to some current T , and we write Tl  T , if Tl (ω) → T (ω) ∀ ω ∈ D m . The Plateau problem has now a straightforward generalization in this context of integer rectiVable currents. Generalized Plateau problem. Let  be a compactly supported (m − 1)-dimensional integer rectiVable current in Rm+n with ∂ = 0. Find an m-dimensional integer rectiVable current T such that ∂ T =  and M(T ) ≤ M(S), for every S integer rectiVable with ∂ S = . The success of the theory of integer rectiVable currents is linked ultimately to the possibility to solve the generalized Plateau problem, due to the closure theorem by H. Federer and W. Fleming proven in their pioneering paper [19]. Theorem 1.6 (Federer and Fleming [19]). Let (Tl )l∈N be a sequence of m-dimensional integer rectiFable currents in Rm+n with   sup M(Tl ) + M(∂ Tl ) < +∞, l∈N

and assume that Tl  T . Then, T is an integer rectiFable current. It is then natural to ask about the regularity properties of the solutions to the generalized Plateau problem, called in the sequel area minimizing integer rectiVable currents.

135 Higher codimension integral currents

1.2 Partial regularity in higher codimension The regularity theory for area minimizing integer rectiVable currents depends very much on the dimension of the current and its codimension in the ambient space (i.e., using the same letters as above, if T is an mdimensional current in Rm+n , the codimension is n). In this course we are interested in the general case of currents with higher codimensions n > 1. The case n = 1 is usually treated separately, because different techniques can be used and more reVned results can be proven (see [10,20,28,30,32,33] for the interior regularity and [3,23] for the boundary regularity). In higher codimension the most general result is due to F. Almgren [5] and concerns the interior partial regularity up to a (relatively) closed set of dimension at most m − 2. Theorem 1.7 (Almgren [5]). Let T be an m-dimensional area minimizing integer rectiFable current in Rm+n . Then, there exists a closed set Sing(T) of Hausdorff dimension at most m − 2 such that in Rm+n \ (spt (∂ T ) ∪ Sing(T)) the current T is induced by the integration over a smooth oriented submanifold of Rm+n . In the next pages I will give an overview of the new proof of Theorem 1.7 given in collaboration with C. De Lellis in a series of papers [13–17]. Although our proof is considerably simpler than the original one, it remains quite involved: this text is, therefore, meant as a survey of the techniques and the various steps of the proof, and can be considered an introduction to the reading of the papers [14, 15, 17]. Remark 1.8. The interior partial regularity can be proven for integer rectiVable currents in a Riemannian manifold M. In [5] Almgren proves the result for C 5 regular ambient manifolds M, while our papers [14, 15, 17] extend this result to C 3,α regular manifolds. Further notation and terminology Given an m-dimensional integer rectiVable current T = (R, τ, θ), we shall often use the following standard notation: T  := |θ| Hm

R,

T := τ

and spt (T ) := spt (T ).

The regular and the singular part of a current are deVned as follows.  Reg(T) := x ∈ spt (T) : spt (T) ∩ Br (x) is induced by a smooth

 submanifold for some r > 0 ,   Sing(T) := spt (T) \ spt (∂T) ∪ Reg(T) .

136 Emanuele Spadaro

2 The blowup argument: a glimpse of the proof The main idea of the proof of Theorem 1.7 is to detect the singularities of an area minimizing current by a blowup analysis. For any r > 0 and x ∈ Rm+n , let ιx,r denote the map ιx,r : y →

y−x , r

and set Tx,r := (ιx,r )" T , where " is the push-forward operator, namely (ιx,r )" T (ω) := T (ι∗x,r ω) ∀ ω ∈ D m . By the classical monotonicity formula (see, e.g., [2, Section 5]), for every rk ↓ 0 and x ∈ spt (T ) \ spt (∂ T ), there exists a subsequence (not relabeled) such that Tx,rk  S, where S is a cone without boundary (i.e. S0,r = S for all r > 0 and ∂ S = 0) which is locally area minimizing in Rm+n . Such a cone will be called, as usual, a tangent cone to T at x. The idea of the blowup analysis dates back to De Giorgi’s pioneering paper [10] and has been used in the context of codimension one currents to recognize singular points and regular points, because in this case the tangent cones to singular and regular points are in fact different. 2.1 Flat tangent cones do not imply regularity This is not the case for higher codimension currents. In order to illustrate this point, let us consider the current TV induced by the complex curve considered above:   V = (z, w) : z 3 = w2 , |z| ≤ 1 ⊂ C2  R4 . It is simple to show that TV is an area minimizing integer rectiVable current (cp. [18, 5.4.19]), which is singular in the origin. Nevertheless, the unique tangent cone to TV at 0 is the current S = (R2 × {0}, e1 ∧ e2 , 2) which is associated to the integration on the horizontal plane R2 × {0}  {w = 0} with multiplicity two. The tangent cone is actually regular, although the origin is a singular point! 2.2 Non-homogeneous blowup One of the main ideas by Almgren is then to extend this reasoning to different types of blowups, by rescaling differently the “horizontal directions”, namely those of a Wat tangent cone at the point, and the “vertical”

137 Higher codimension integral currents

ones, which are the orthogonal complement to the former. In this way, in place of preserving the geometric properties of the rectiVable current T , one is led to preserve the energy of the associated multiple valued function. In order to explain this point, let us consider again the current TV . The support of such current, namely the complex curve V, can be viewed as the graph of a function which associates to any z ∈ C with |z| ≤ 1 two points in the w-plane: z → {w1 (z), w2 (z)} with wi (z)2 = z 3 for i = 1, 2.

(2.1)

Then the right rescaling according to Almgren is the one producing in the limit a multiple valued harmonic function preserving the Dirichlet energy (for the deVnitions see the next sections). In the case of V, the correct rescaling is the one Vxing V. For every λ > 0, we consider λ : C2 → C2 given by λ (z, w) = (λ2 z, λ3 w), and note that (λ )" TV = TV for every λ > 0. Indeed, in the case of V the functions w1 and w2 , being the two determinations of the square root of z 3 , are already harmonic functions (at least away from the origin). 2.3 Multiple valued functions Following these arguments, we have then to face the problem of deVning harmonic multiple valued functions, and to study their singularities. Abstracting from the above example, we consider the multiple valued functions from a domain in Rm which take a Vxed number Q ∈ N \ {0} of values in Rn . This functions will be called in the sequel Q-valued functions. The deVnition of harmonic Q-valued functions is a simple issue around any “regular point” x0 ∈ Rm , for it is enough to consider just the superposition of classical harmonic functions (possibly with a constant integer multiplicity), i.e. Rm ⊃ Br (x0 ) # x → {u 1 (x), . . . , u Q (x)} ∈ (Rn ) Q ,

(2.2)

with u i harmonic and either u i = u j or u i (x)  = u i (x) for every x ∈ Br (x0 ). The issue becomes much more subtle around the singular points. As it is clear from the example (2.1), in a neighborhood of the origin there is no representation of the map z → {w1 (z), w2 (z)} as in (2.2). In this case the two values w1 (z) and w2 (z) cannot be ordered in a consistent way

138 Emanuele Spadaro

(due to the branch point at 0), and hence cannot be distinguished one from the other. We are then led to consider a multiple valued function as a map taking Q values in the quotient space (Rn ) Q / ∼ induced by the symmetric group S Q of permutation of Q indices: namely, given points Pi , Si ∈ Rn , (P1 , . . . , PQ ) ∼ (S1 , . . . , S Q ) if there exists σ ∈ S Q such that Pi = Sσ (i) for every i = 1, . . . , Q. Note that the space (Rn ) Q / ∼ is a singular metric space (for a naturally deVned metric, see the next section). Therefore, harmonic maps with values in (Rn ) Q / ∼ have to be carefully deVned, for instance by using the metric theory of harmonic functions developed in [22,25,26] (cp. also [13, 27]). Remark 2.1. Note that the integer rectiVable current induced by the graph of a Q-valued function (under suitable hypotheses, cp. [16, Proposition 1.4]) belongs to a subclass of currents, sometimes called “positively oriented”, i.e. such that the tangent planes make at almost every point a positive angle with a Vxed plane. Nevertheless, as it will become clear along the proof, it is enough to consider this subclass as model currents in order to conclude Theorem 1.7. 2.4 The need of centering A major geometric and analytic problem has to be addressed in the blowup procedure sketched above. In order to make it apparent, let us discuss another example. Consider the complex curve W given by   W = (z, w) : (w − z 2 )2 = z 5 , |z| ≤ 1 ⊂ C2 . As before, W can be associated to an area minimizing integer rectiVable current TW in R4 , which is singular at the origin. It is easy to prove that the unique tangent plane to TW at 0 is the plane {w = 0} taken with multiplicity two. On the other hand, by simple analytical considerations, the only nontrivial inhomogeneous blowup in these vertical and horizontal coordinates is given by λ (z, w) = (λ z, λ2 w), and (λ )" TW converges as λ → +∞ to the current induced by the smooth complex curve {w = z 2 } taken with multiplicity two. In other words, the inhomogeneous blowup did not produce in the limit any singular current and cannot be used to study the singularities of TW . For this reason it is essential to “renormalize” TW by averaging out its regular Vrst expansion, on top of which the singular branching behavior

139 Higher codimension integral currents

happens. In the case we handle, the regular part of TW is exactly the smooth complex curve {w = z 2 }, while the singular branching is due to the determinations of the square root of z 5 . It is then clear why one can look for parametrizations of W deVned in {w = z 2 }, so that the singular map to be considered reduces to z → {u 1 (z), u 2 (z)} with u 1 (z)2 = z 5 . The regular surface {w = z 2 } is called center manifold by Almgren, because it behaves like (and in this case it is exactly) the average of the sheets of the current in a suitable system of coordinates. In general the determination of the center manifold is not straightforward as in the above example, and actually constitutes the most intricate part of the proof. 2.5 Excluding an inHnite order of contact Having taken care of the geometric problem of the averaging, in order to be able to perform successfully the inhomogeneous blowup, one has to be sure that the Vrst singular expansion of the current around its regular part does not occur with an inVnite order of contact, because in that case the blowup would be by necessity zero. This issue involves one of the most interesting and original ideas of F. Almgren, namely a new monotonicity formula for the so called frequency function (which is a suitable ratio between the energy and a zero degree norm of the function parametrizing the current). This is in fact the right monotone quantity for the inhomogeneous blowups introduced before, and it allows to show that the Vrst singular term in the “expansion” of the current does not occur with inVnite order of contact and actually leads to a nontrivial limiting current. 2.6 The persistence of singularities Finally, in order to conclude the proof we need to assure that the singularities of the current do transfer to singularities of the limiting multiple valued function, which can be studied with more elementary techniques. This is in general not true in a pointwise sense, but it becomes true in a measure theoretic sense as soon as the singular set is supposed to have positive Hm−2+α measure, for some α > 0. The contradiction is then reached in the following way: starting from an area minimizing current with a big singular set (Hm−2+α positive measure), one can perform the analysis outlined before and will end up with a multiple valued function having a big set of singularities, thus giving the desired contradiction.

140 Emanuele Spadaro

2.7 Sketch of the proof The rigorous proof of Theorem 1.7 is actually much more involved and complicated than the rough outline given in the previous section, and can be found either in [5] or in the recent series of papers [13–17]. In this lecture notes we give some more details of this recent new proof, and comments on some of the subtleties which were hidden in the general discussion above. Since the proof is very lengthly, we start with a description of the strategy. The proof is done by contradiction. We will, indeed, always assume the following in the sequel. Contradiction assumption: there exist numbers m ≥ 2, n ≥ 1, α > 0 and an area minimizing m-dimensional integer rectiVable current T in Rm+n such that Hm−2+α (Sing(T)) > 0. Note that the hypothesis m ≥ 2 is justiVed because, for m = 1 an area minimizing current is locally the union of Vnitely many non-intersecting open segments. The aim of the proof is now to show that there exist suitable points of Sing(T) where we can perform the blowup analysis outlined in the previous section. This process consists of different steps, which we next list in a way which does not require the introduction of new notation but needs to be further speciVed later. (A) Find a point x0 ∈ Sing(T) and a sequence of radii (rk )k with rk ↓ 0 such that: (A1 ) the rescaling currents Tx0 ,rk := (ιx0 ,rk )" T converge to a Wat tangent cone; (A2 ) Hm−2+α (Sing(Tx0 ,rk ) ∩ B1 ) > η > 0 for some η > 0 and for every k ∈ N. Note that both conclusions hold for suitable subsequences, which in principle may not coincide. What we need to prove is that we can select a point and a subsequence satisfying both. (B) Construction of the center manifold M and of a normal Lipschitz approximation N : M → Rm+n / ∼. This is the most technical part of the proof, and most of the conclusions of the next steps will intimately depend on this construction. (C) The center manifold that one constructs in step (B) can only be used in general for a Vnite number of radii rk of step (A). The reason is that

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in general its degree of approximation of the average of the minimizing currents T is under control only up to a certain distance from the singular point under consideration. This leads us to deVne the sets where the approximation works, called in the sequel intervals of Gattening, and to deVne an entire sequence of center manifolds which will be used in the blowup analysis. (D) Next we will take care of the problem of the inVnite order of contact. This is done in two part. For the Vrst one we derive the almost monotonicity formula for a variant of Almgren’s frequency function, deducing that the order of contact remains Vnite within each center manifold of the sequence in (C). (E) Then one needs to compare different center manifolds and to show that the order of contact still remains Vnite. This is done by exploiting a deep consequence of the construction in (C) which we call splitting before tilting after the inspiring paper by T. Rivi`ere [29]. (F) With this analysis at hand, we can pass into the limit our blowup sequence and conclude the convergence to the graph of a harmonic Qvalued function u. (G) Finally, we discuss the capacitary argument leading to the persistence of the singularities, to show that the function u in (F) needs to have a singular set with positive Hm−2+α measure, thus contradicting the partial regularity estimate for such multiple valued harmonic functions. In the remaining part of this course we give a more detailed description of the steps above, referring to the original papers [13–17] for the complete proofs.

3 Q-valued functions and rectiHable currents Since the Vnal contradiction argument relies on the regularity theory of multiple valued functions, we start recalling the main deVnitions and results concerning them, and the way they can be used to approximate integer rectiVable currents. The reference for this part of the theory is [13, 16, 17, 34]. 3.1 Q-valued functions We start by giving a metric structure to the space (Rn ) Q / ∼ of unordered Q-tuples of points in Rn , where Q ∈ N \ {0} is a Vxed number. It is immediate to see that this space can be identiVed with the subset of positive measures of mass Q which are the sum of integer multiplicity Dirac

142 Emanuele Spadaro

delta: (Rn ) Q / ∼ 

( Q 

A Q (Rn ) :=

) [[Pi ]] : Pi ∈ Rn ,

i=1

where [[Pi ]] denotes the Dirac delta at Pi . We can then endow A Q with one of the distances deVned for (probability) measures, for example the = Wasserstein distance of exponent two: for every T 1 i [[Pi ]] and T2 =  n ∈ A (R ), we set [ [S ] ] i Q i . / Q /    Pi − Sσ (i) 2 , G(T1 , T2 ) := min 0 σ ∈S Q

i=1

where we recall that S Q denotes the symmetric group of Q elements. A Q-function simply a map f :  → A Q (Rn ), where  ⊂ Rm is an open domain. We can then talk about measurable (with respect to the Borel σ -algebra of A Q (Rn )), bounded, uniformly-, H¨older- or Lipschitzcontinuous Q-valued functions. More importantly, following the pioneering approach to weakly differentiable functions with values in a metric space by L. Ambrosio [6], we can also deVne the class of Sobolev Q-valued functions W 1,2 . DeHnition 3.1 (Sobolev Q-valued functions). Let  ⊂Rm be a bounded open set. A measurable function f :  → A Q is in the Sobolev class W 1,2 if there exist m functions ϕ j ∈ L 2 () for j = 1, . . . , m, such that 1,2 (i) x → G( f (x),  T ) ∈ W () for all T ∈ A Q ; (ii) ∂ j G( f, T ) ≤ ϕ j almost everywhere in  for all T ∈ A Q and for all j ∈ {1, . . . , m}, where ∂ j G( f, T ) denotes the weak partial derivatives of the functions in (i).

By simple reasonings, one can infer the existence of minimal functions |∂ j f | fulVlling (ii): |∂ j f | ≤ ϕ j a.e. for any other ϕ j satisfying (ii), We set |D f |2 :=

m    ∂ j f 2 ,

(3.1)

j=1

and deVne the Dirichlet energy of a Q-valued function as (cp. also [25– 27] for alternative deVnitions)  |D f |2 . Dir( f ) := 

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A Q-valued function f is said Dir-minimizing if   2 |D f | ≤ |Dg|2 for all g ∈ W

 1,2



(3.2)

(, A Q ) with G( f, g)|∂ = 0,

where the last inequality is meant in the sense of traces. The main result in the theory of Q-valued functions is the following. Theorem 3.2. Let  ⊂ Rm be a bounded open domain with Lipschitz boundary, and let g ∈ W 1,2 (, A Q (Rn )) be Fxed. Then, the following holds. (i) There exists a Dir-minimizing function f solving the minimization problem (3.2). 0,κ (, A Q (Rn )) for a dimen(ii) Every such function f belongs to Cloc sional constant κ = κ(m, Q) > 0. p (iii) For every such function f , |D f | ∈ L loc () for some dimensional constant p = p(m, n, Q) > 2. (iv) There exists a relatively closed set Sing(u) ⊂  of Hausdorff dimension at most m − 2 such that the graph of u outside Sing(u), i.e. the set graph u|\ = {(x, y) : x ∈  \ , y ∈ spt (u(x))} , is a smoothly embedded m-dimensional submanifold of Rm+n . Remark 3.3. We refer to [13, 34] for the proofs and more reVned results in the case of of two dimensional domains. Moreover, for some results concerning the boundary regularity we refer to [24], and for an improved estimate of the singular set to [21]. We close this section by some considerations on the Q-valued functions. For the reasons explained in the previous section, a Q-valued function has to be considered as an intrinsic map taking values in the non-smooth space of Q-points A Q , and cannot be reduced to a “superposition” of a number Q of functions. Nevertheless, in many situations it is possible to handle Q-valued functions as a superposition. For example, as shown in [13, Proposition 0.4] every measurable function f : Rm → A Q (Rn ) can be written (not uniquely!) as f (x) =

Q 

[[ f i (x)]]

for Hm -a.e. x,

i=1

with f 1 , . . . , f Q : Rm → Rn measurable functions.

(3.3)

144 Emanuele Spadaro

Similarly, for weakly differentiable functions it is possible to deVne a notion of pointwise approximate differential (cp. [13, Corollary 2,7])  Df = [[D f i ]] ∈ A Q (Rn×m ), i

with the property that at almost every x it holds D f i (x) = D f j (x) if f i (x) = f j (x). Note, however, that the functions f i do not need to be weakly differentiable in (3.3), for the Q-valued function f has an approximate differential. 3.2 Graph of Lipschitz Q-valued functions There is a canonical way to give the structure of integer rectiVable currents to the graph of a Lipschitz Q-valued function. To this aim, we consider proper Q-valued functions, i.e. measurable functions F : M → A Q (Rm+n ) (where M is any m-dimensional  submanifold of Rm+n ) such that there is a measurable selection F = i [[Fi ]] for which (Fi )−1 (K ) i

is compact for every compact K ⊂ Rm+n . It is then obvious that if there exists such a selection, then every measurable selection shares the same property. By a simple induction argument (cp. [16, Lemma 1.1]), there are a countable partition of M in bounded measurable subsets Mi (i ∈ N) and j Lipschitz functions f i : Mi → Rm+n ( j ∈ {1, . . . , Q}) such that (a) F| Mi =

 Q  j=1

fi

j



j

for every i ∈ N and Lip( f i ) ≤ Lip(F) ∀i, j; j

j

j

j

(b) ∀ i ∈ N and j, j  ∈ {1, . . . , Q}, either f i ≡ f i or f i (x)  = f i (x) ∀x ∈ Mi ;    j (c) ∀ i we have D F(x) = Qj=1 D f i (x) for a.e. x ∈ Mi . We can then give the following deVnition. DeHnition 3.4 (Q-valued push-forward). Let M be an oriented submanifold of Rm+n of dimension m and let F : M → A Q (Rm+n ) be a proper Lipschitz map. Then, we deVne the push-forward of M through F as the current  j ( f i )" [[Mi ]] , TF = i, j

145 Higher codimension integral currents

j

where Mi and f i are as above: that is, T F (ω) :=

Q  

j

j

ω( f i (x)), D f i (x)" e(x)  dHm (x) ∀ ω ∈ D m (Rn ).

i∈N j=1 Mi

(3.4)

One can prove that the current in DeVnition 3.4 does not depend on the decomposition chosen for M and f , and moreover is integer rectiVable (cp. [16, Proposition 1.4]) A particular class of push-forwards are given by graphs.  DeHnition 3.5 (Q-graphs). Let f = i [[ f i ]] : Rm→A Q (Rn ) be Lipschitz Q and deVne the map F : M → A Q (Rm+n ) as F(x) := i=1 [[(x, f i (x))]]. Then, T F is the current associated to the graph Gr( f ) and will be denoted by G f . The main result concerning the push-forward of a Q-valued function is the following (see [16, Theorem 2.1]). Theorem 3.6 (Boundary of the push-forward). Let M ⊂ Rm+n be an m-dimensional submanifold with boundary, F : M → A Q (Rm+n ) a proper Lipschitz function and f = F|∂ M . Then, ∂T F = T f . Moreover, the following Taylor expansion of the mass of a graph holds (cp. [16, Corollary 3.3]). Proposition 3.7 (Expansion of M(G f )). There exist dimensional constants c, ¯ C > 0 such that, if  ⊂ Rm is a bounded open set and f : ¯ then  → A Q (Rn ) is a Lipschitz map with Lip( f ) ≤ c,    1 2 M(G f ) = Q|| + (3.5) |D f | + R¯ 4 (D f i ) , 2   i ¯ where R¯ 4 ∈ C 1 (Rn×m ) satisFes | R¯ 4 (D)| = |D|3 L(D) for L¯ : Rn×m → R ¯ ≤ C and L(0) ¯ Lipschitz with Lip( L) = 0. 3.3 Approximation of area minimizing currents Finally we recall some results on the approximation of area minimizing currents. To this aim we need to introduce more notation. We consider cylinders in Rm+n of the form C¯ s (x) := B¯ s (x) × Rn with x ∈ Rm . Since we are interested in interior regularity, we can assume for the purposes of this section that we are always in the following setting: for

146 Emanuele Spadaro

some open cylinder C¯ 4r (x) (with r ≤ 1) and some positive integer Q, the area minimizing current T has compact support in C¯ 4r (x) and satisVes   and ∂ T C¯ 4r (x) = 0, (3.6) p" T = Q B¯ 4r (x) where p : Rm+n → π0 := Rm × {0} is the orthogonal projection. We introduce next the main regularity parameter for area minimizing currents, namely the Excess. DeHnition 3.8 (Excess measure). For a current T as above we deVne the cylindrical excess E(T, C¯ r (x)) as follows: T (C¯ r (x)) E(T, C¯ r (x)) := −Q ωm r m  1 |T − π0 |2 δT , = 2 ωm r m T (C¯ r (x)) where ωm is the measure of the m-dimensional unit ball, and π0 is the m-vector orienting π0 . The most general approximation result of area minimizing currents is the one due to Almgren, and reproved in [17] with more reVned techniques, which asserts that under suitable smallness condition of the excess, an area minimizing current coincides on a big set with a graph of a Lipschitz Q-valued function. Theorem 3.9 (Almgren’s strong approximation). There exist constants C, γ1 , ε1 > 0 (depending on m, n, Q) with the following property. Assume that T is area minimizing in the cylinder C¯ 4r (x) and assume that E := E(T, C¯ 4 r (x)) < ε1 . Then, there exist a map f : Br (x) → A Q (Rn ) and a closed set K ⊂ B¯ r (x) such that the following holds: Lip( f ) ≤ C E γ1 ,

(3.7)

r , (3.8) G f (K ×R ) = T (K × R ) and |Br (x)\ K | ≤ C E      T (C¯ r (x)) − Q ωm r m − 1 |D f |2  ≤ C E 1+γ1 r m . (3.9)  2 n

n

1+γ1

m

Br (x)

The most important improvement of the theorem above with respect to the preexisting approximation results is the small power E γ1 in the three estimates (3.7) - (3.9). Indeed, this will play a crucial role in the construction of the center manifold. It is worthy mentioning that, when Q = 1

147 Higher codimension integral currents

and n = 1, this approximation theorem was Vrst proved with different techniques by De Giorgi in [10] (cp. also [12, Appendix]). As a byproduct of this approximation, we also obtain the analog of the so called harmonic approximation, which allows us to compare the Lipschitz approximation above with a Dir-minimizing function. Theorem 3.10 (Harmonic approximation). Let γ1 , ε1 be the constants of Theorem 3.9. Then, for every η¯ > 0, there is a positive constant ε¯ 1 < ε1 with the following property. Assume that T is as in Theorem 3.9 and E := E(T, C¯ 4 r (x)) < ε¯ 1 . If f is the map in Theorem 3.9, then there exists a Dir-minimizing function w such that r −2



 G( f, w)2 + Br (x)  +

Br (x)

Br (x)

(|D f | − |Dw|)2

|D(η ◦ f ) − D(η ◦ w)|2 ≤ η¯ E r m , (3.10)

where η : A Q (Rn ) → Rn is the average map η

1  i

2 [[Pi ]] =

1  Pi . Q i

4 Selection of contradiction’s sequence In this section we give the details of the Vrst step (A) in Section 2.7, namely the selection of a common subsequence such that the rescaled currents converge to a Wat tangent cone and the measure of the singular set remains uniformly bounded below away from zero. For this purpose, we introduce the following notation. We denote by Br (x) the open ball of radius r > 0 in Rm+n (we do not write the point x if the origin) and, for Q ∈ N, we denote by D Q (T ) the points of density Q of the current T , and set RegQ (T) := Reg(T) ∩ DQ (T) and SingQ (T) := Sing(T) ∩ DQ (T). The precise properties of the sequence that will be used in the blowup argument are stated in the following proposition. We recall that the main hypothesis at the base of the proof is the contradiction assumption of Section 2.7, which we restate for reader’s convenience.

148 Emanuele Spadaro

Contradiction assumption: there exist numbers m ≥ 2, n ∈ N, α > 0 and an area minimizing m-dimensional integer rectiVable current T in Rm+n such that Hm−2+α (Sing(T)) > 0. We introduce the spherical excess deVned as follows: for a given mdimensional plane π,  1 |T − π|2 dT , E(T, Br (x), π) := 2 ωm r m Br (x) E(T, Br (x)) := min E(T, Br (x), τ ). τ

Proposition 4.1 (Contradiction’s sequence). Under the contradiction assumption, there exist 1. constants m, n, Q ≥ 2 natural numbers and α, η > 0 real numbers; 2. an m-dimensional area minimizing integer rectiFable current T in Rm+n with ∂ T = 0; 3. a sequence rk ↓ 0 such that 0 ∈ D Q (T ) and the following holds:

lim

k→+∞

lim E(T0,rk , B10 ) = 0, k→+∞ m−2+α H∞ (D Q (T0,rk ) ∩ B1 )

(4.1) > η,

  Hm (B1 ∩ spt (T0,rk )) \ D Q (T0,rk ) > 0 ∀ k ∈ N.

(4.2) (4.3)

m−2+α is the Hausdorff premeasure computed without any restricHere H∞ tion on the diameter of the sets in the coverings. By Almgren’s stratiVcation theorem and by general measure theoretic arguments, there exist sequences satisfying either (4.1) or (4.2). The two subsequences might, however, be different: we show the existence of one point and a single subsequence along which both conclusions hold. The proof of the proposition is based on the following two results.

Theorem 4.2 (Almgren [5, 2.27]). Let α > 0 and let T be an integer rectiFable area minimizing current in Rm+n . Then, (1) for Hm−2+α -a.e. point x ∈ spt (T ) \ spt (∂ T ) there exists a subsequence sk ↓ 0 such that Tx,sk converges to a Gat cone; (2) for Hm−3+α -a.e. point x ∈ spt (T )\spt (∂ T ), it holds that !(T, x) ∈ Z.

149 Higher codimension integral currents

Lemma 4.3. Let S be an m-dimensional area minimizing integral current, which is a cone in Rm+n with ∂ S = 0, Q = !(S, 0) ∈ N \ {0}, and assume that  Hm D Q (S)) > 0 and Hm−1 (SingQ (S)) = 0. Then S is an m-dimensional plane with multiplicity Q. Proof of Proposition 4.1. Let m > 1 be the smallest integer for which Theorem 1.7 fails. In view of Almgren’s stratiVcation Theorem 4.2, we can assume that there exist an integer rectiVable area minimizing current R of dimension m and a positive integer Q such that the Hausdorff dimension of SingQ (R) is larger than m − 2. We Vx the smallest Q for which such a current R exists and note that by Allard’s regularity theorem (cp. [2]) it must be Q > 1. Let α > 0 be such that Hm−2+α (SingQ (R)) > 0, and consider a density point x0 for the measure Hm−2+α (without loss of generality x0 = 0). In particular, there exists rk ↓ 0 such that   m−2+α SingQ (R) ∩ Brk H∞ > 0. lim k→+∞ rkm−2+α Up to a subsequence (not relabeled) we can assume that R0,rk → S, with S a tangent cone. If S is a multiplicity Q Wat plane, then we set T := R and the proposition is proven (indeed, (4.3) is satisVed because 0 ∈ Sing(R) and R ≥ Hm spt (R)). If S is not Wat, taking into account the convergence properties of area minimizing currents [31, Theorem 34.5] and the upper semicontinuity of m−2+α under the Hausdorff convergence of compact sets, we deduce H∞     m−2+α m−2+α ¯ 1 ≥ lim inf H∞ ¯ 1 > 0. (4.4) D Q (S) ∩ B D Q (R0,rk ) ∩ B H∞ k→+∞

We claim that (4.4) implies m−2+α (SingQ (S)) > 0. H∞

(4.5)

Indeed, if all points of D Q (S) are singular, then (4.5) follows from (4.4) directly. Otherwise, RegQ (S) is not empty, thus implying Hm (D Q (S) ∩ B1 ) > 0: we can then apply Lemma 4.3 and infer that, since S is not regular, then Hm−1 (SingQ (S)) > 0 and (4.5) holds. We can, hence, Vnd x ∈ SingQ (S) \ {0} and rk ↓ 0 such that  m−2+α SingQ (S) ∩ Brk (x)) H∞ lim > 0. k→+∞ rkm−2+α

150 Emanuele Spadaro

Up to a subsequence (not relabelled), we can assume that Sx,rk converges to S1 . Since S1 is a tangent cone to the cone S at x  = 0, S1 splits off a line, i.e. S1 = S2 × [[{t e : t ∈ R}]] for some e ∈ Sm+n−1 , for some area minimizing cone S2 in Rm−1+n and some v ∈ Rm+n (cp. [31, Lemma 35.5]). Since m is, by assumption, the smallest integer for which Theorem 1.7 fails, Hm−3+α (Sing(S2 )) = 0 and, hence, Hm−2+α (SingQ (S1 )) = 0. On the other hand, arguing as for (4.4), we have m−2+α m−2+α ¯ 1 ) ≥ lim sup H∞ ¯ 1 ) > 0. (D Q (S1 ) ∩ B (D Q (Sx,rk ) ∩ B H∞ k→+∞

Thus RegQ (S1 )  = ∅ and, hence, Hm (D Q (S1 )) > 0. We then can apply Lemma 4.3 again and conclude that S1 is an m-dimensional plane with multiplicity Q. Therefore, the proposition follows taking T a suitable translation of S. Proof of Lemma 4.3 We premise the following lemma. Lemma 4.4. Let T be an integer rectiFable current of dimension m in Rm+n with locally Fnite mass and U an open set such that Hm−1 (∂U ∩ spt (T )) = 0 and (∂ T ) U = 0. Then ∂(T

U ) = 0.

Proof. Consider V ⊂⊂ Rm+n . By the slicing theory Sr := T (V ∩ U ∩ {dist (x, ∂U ) > r}) is a normal current in Nm (V ) for a.e. r. Since M(T (V ∩ U ) − Sr ) → 0 as r ↓ 0, we conclude that T (U ∩V ) is in the M-closure of Nm (V ). Thus, by [18, 4.1.17], T U is a Wat chain in Rm+n and by [18, 4.1.12] ∂(T U ) is a Wat chain. Since spt (∂(T U )) ⊂ ∂U ∩ spt (T ), we can apply [18, Theorem 4.1.20] to conclude that ∂(T U ) = 0. We next prove Lemma 4.3. For each x ∈ RegQ (S), let r x be such that S B2rx (x) = Q [[]] for some regular submanifold  and set Brx (x).

U := x∈RegQ (S)

151 Higher codimension integral currents

Obviously, RegQ (S) ⊂ U; hence, by assumption, it is not empty. Fix x ∈ spt (S) ∩ ∂U . Let next (xk )k∈N ⊂ RegQ (S) be such that dist (x, Brxk (xk )) → 0. We necessarily have that r xk → 0: otherwise we would have x ∈ B2rxk (xk ) for some k, which would imply x ∈ RegQ (S) ⊂ U, i.e. a contradiction. Therefore, xk → x and, by [31, Theorem 35.1], Q = lim sup !(S, xk ) ≤ !(S, x) = lim !(S, λx) ≤ !(S, 0) = Q. k→+∞

λ↓0

This implies x ∈ D Q (S). Since x ∈ ∂U , we must then have x ∈ SingQ (S). Thus, we conclude that Hm−1 (spt (S) ∩ ∂U ) = 0. It follows from Lemma 4.4 that S  := S U has 0 boundary in Rm+n . Moreover, since S is an area minimizing cone, S  is also an area-minimizing cone. By deVnition of U we have !(S  , x) = Q for S  -a.e. x and, by semicontinuity, Q ≤ !(S  , 0) ≤ !(S, 0) = Q. We apply Allard’s theorem [2] and deduce that S  is regular, i.e. S  is an m-plane with multiplicity Q. Finally, from !(S  , 0) = !(S, 0), we infer S  = S.

5 Center manifold’s construction In this section we describe the procedure for the construction of the center manifold. As mentioned in the introduction, this is the most complicated part of the proof: indeed, the construction of the center manifold comes together with a series of other estimates which will enter signiVcantly in the proof of the main Theorem 1.7. In particular, as an outcome of the procedure we obtain the following several things. (1) A decomposition of the horizontal plane π0 = Rm × {0} of “Whitney’s type”. (2) A family of interpolating functions deVned on the cubes of this decomposition. (3) A normal approximation taking values in the normal bundle of the center manifold. (4) A set of criteria (which will in fact determine the Whitney decomposition) which lead to what we call splitting-before-tilting estimates. (5) An family of intervals, called intervals of Gattening, where the construction will be effective. (6) A family of pairs cube–ball transforming the estimates on the Whitney decomposition into estimates on balls (thus passing from the cubic lattice of the decomposition to the standard geometry of balls).

152 Emanuele Spadaro

5.1 Notation and assumptions Let us recall the following notation. Given an integer rectiVable current T with compact support, we consider the spherical and the cylindrical excesses deVned as follows, respectively: for given m-planes π, π  , we set    m −1 E(T, Br (x), π) := 2ωm r |T − π|2 dT , (5.1) Br (x)

−1  E(T, C¯ r (x, π), π  ) := 2ωm r m



C¯ r (x,π)

|T − π  |2 dT  , (5.2)

where C¯ r (x, π) = B¯ r (x, π) × π ⊥ is the cylinder over the closed ball B¯ r (x, π) or radius r and center x in the m-dimensional plane π. And we consider the height function in a set A (we denote by pπ the orthogonal projection on a plane π) h(T, A, π) :=

sup

x,y ∈ spt (T ) ∩ A

|pπ ⊥ (x) − pπ ⊥ (y)| .

We also set E(T, Br (x)) := min E(T, Br (x), τ ) = E(T, Br (x), π), τ

(5.3)

and we will use E(T, C¯ r (x, π)) in place of E(T, C¯ r (x, π), π): note that it coincides with the cylindrical excess as deVned in Section 3.3 when   (pπ )" T C¯ r (x, π) = Q B¯ r (pπ (x), π) . In this section we will work with an area minimizing integer rectiVable current T 0 with compact support which satisVes the following assumptions: for some constant ε2 ∈ (0, 1), which we always suppose to be small enough, !(0, T 0 ) = Q and ∂ T 0 B6√m = 0,   √ T 0 (B6√mρ ) ≤ ωm Q(6 m)m + ε22 ρ m ∀ρ ≤ 1,     E := E T 0 , B6√m = E T 0 , B6√m , π0 ≤ ε22 ,

(5.4) (5.5) (5.6)

It follows from standard considerations in geometric measure theory that there are positive constants C0 (m, n, Q) and c0 (m, n, Q) with the following property. If T 0 is as in (5.4) - (5.6), ε2 < c0 and T := T 0 B23√m/4 ,

153 Higher codimension integral currents

then: C¯ 11√m/2 (0, π0 ) = 0,   C¯ 11√m/2 (0, π0 ) = Q B11√m/2 (0, π0 ) , ∂T

(pπ0 )" T

1 m

h(T, C¯ 5√m (0, π0 )) ≤ C0 ε2 .

(5.7) (5.8) (5.9)

In particular for each x ∈ B11√m/2 (0, π0 ) there is a point p ∈ spt (T ) with pπ0 ( p) = x. 5.2 Whitney decomposition and interpolating functions The construction of the center manifold is done by following a suitable decomposition of the horizontal plane π0 into cubes. We denote by C j , j ∈ N, the family of dyadic-closed cubes L of π0 with side-length 21− j =: 2 &(L). Next we set C := j∈N C j . If H and L are two cubes in C with H ⊂ L, then we call L an ancestor of H and H a descendant of L. When in addition &(L) = 2&(H ), H is a son of L and L the father of H . DeHnition 5.1. A Whitney decomposition of [−4, 4]m ⊂ π0 consists of a closed set  ⊂ [−4, 4]m and a family W ⊂ C satisfying the following properties: (w1)  ∪ L∈W L = [−4, 4]m and  does not intersect any element of W; (w2) the interiors of any pair of distinct cubes L 1 , L 2 ∈ W are disjoint; (w3) if L 1 , L 2 ∈ W have nonempty intersection, then 1 &(L 1 ) ≤ &(L 2 ) ≤ 2 &(L 1 ). 2 Observe that (w1) - (w3) imply   dist (, L) := inf |x − y| : x ∈ L , y ∈  ≥ 2&(L) for every L ∈ W. However, we do not require any inequality of the form dist (, L) ≤ C&(L), although this would be customary for what is commonly called Whitney decomposition in the literature. We denote by S j all the dyadic cubes with side-length 21− j which are not contained in W and set S := ∪ j≥N0 S j for some √ big natural number N0 . For each cube L ∈ W ∪ S , we set r L = M0 m&(L), with M0 ∈ N a dimensional constant to be Vxed later, and we call its center x L . We can then Vnd points p L ∈ spt (T ), with coordinates p L = (x L , yL ) ∈ π0 ×π0⊥ , and interpolating functions g L : B4r L ( p L , π0 ) → π0⊥ ,

154 Emanuele Spadaro

such that the following holds: for every H, L ∈ W ∪ S , 1

g H C 0 ≤ C E 2m

1

and Dg H C 2,κ ≤ C E 2 ; 1 2

g H − g L C i (Br L ( p L ,π0 )) ≤ C E &(H )3+κ−i

(5.10) (5.11)

∀ i ∈ {0, . . . , 3} if H ∩ L = ∅; 1

|D 3 g H (x H ) − D 3 g L (x L )| ≤ C E 2 |x H − x L |κ ; sup

(x,y)∈spt (T ), x∈H

g H − yC 0 ≤ C E

1 2m

&(H ),

(5.12) (5.13)

for some κ > 0, and where we used the notation Br ( p L , π0 ) := Br ( p L ) ∩ ( p L + π0 ). It is now very simple to show how to patch all the interpolating functions g L in order to construct a center manifold. To this aim, we set   P j := S j ∪ L ∈ W : &(L) ≥ 2− j . For every L ∈ P j we deVne



 y − xL ϑ L (y) := ϑ , &(L)   , 17 ]m , [0,1] that is identically 1 on [−1,1]m. for some Vxed ϑ ∈ Cc∞ [− 17 16 16 We can then patch all the interpolating functions using the partition of the unit induced by the ϑ L , i.e.  L∈P j ϑ L g L . (5.14) ϕ j :=  L∈P j ϑ L The following theorem is now a very easy consequence of the estimates on the interpolating functions. Theorem 5.2 (Existence of the center manifold). Assume to be given a Whitney decomposition (, W ) and interpolating functions g H as above. If ε2 is sufFciently small, then (i) the functions ϕ j deFned in (5.14) satisfy 1

Dϕ j C 2,κ ≤ C E 2

1

and ϕ j C 0 ≤ C E 2m ,

(ii) ϕ j converges to a map ϕ such that M := Gr(ϕ|]−4,4[m ) is a C 3,κ submanifold of , called in the sequel center manifold,

155 Higher codimension integral currents

(iii) for all x ∈ , the point (x, ϕ(x)) ∈ spt (T ) and is a multiplicity Q point. Setting (y) := (y, ϕ(y)), we call () the contact set.  Proof. DeVne χ H := ϑ H /( L∈P j ϑ L ) and observe that 

χ H = 1 and χ H C i ≤ C0 (i, m, n) &(H )−i

∀i ∈ N . (5.15)

Set P j (H ) := {L ∈ P j : L ∩ H  = ∅} \ {H }. By construction 1 &(L) ≤ &(H ) ≤ 2 &(L) for every L ∈ P j (H ), 2 and the cardinality of P j (H ) is bounded by a geometric constant C0 . 1 The estimate |ϕ j | ≤ C E 2m follows then easily from (5.10). For x ∈ H we write

 ϕ j (x) = g H χ H + g L χ L (x) L∈P j (H )

= g H (x) +



(g L − g H )χ L (x) .

(5.16)

L∈P j (H )

Using the Leibniz rule, (5.15), (5.10) and (5.11), for i ∈ {1, 2, 3} we get   g L − g H C l (H ) &(L)l−i D i ϕ j C 0 (H ) ≤ g H C i + 0≤l≤i L∈P j (H )

  ≤ C E 1 + &(H )3+κ−i . 1 2

1

Next, using also [D 3 g H − D 3 g L ]κ ≤ C E 2 , we obtain [D 3 ϕ j ]κ,H ≤





 &(H )l−3 &(H )−κ Dl (g L − g H )C 0 (H )

0≤l≤3 L∈P j (H )

 1 + [Dl (g L − g H )]κ,H + [D 3 g H ]κ,H ≤ C E 2 .

Fix now x, y ∈ [−4, 4]m , let H, L ∈ P j be such that x ∈ H and y ∈ L. If H ∩ L = ∅, then   |D 3 ϕ j (x) − D 3 ϕ j (y)| ≤ C [D 3 ϕ j ]κ,H + [D 3 ϕ j ]κ,L |x − y|κ . (5.17) If H ∩ L = ∅, we assume without loss of generality &(H ) ≤ &(L) and observe that   max |x − x H |, |y − x L | ≤ &(L) ≤ |x − y| .

156 Emanuele Spadaro

Moreover, by construction ϕ j is identically equal to g H in a neighborhood of its center x H . Thus, we can estimate |D 3 ϕ j (x) − D 3 ϕ j (y)| ≤ |D 3 ϕ j (x) − D 3 ϕ j (x H )| + |D 3 g H (x H ) − D 3 g L (x L )| + |D 3 ϕ j (x L ) − D 3 ϕ j (y)| 1

≤ C E 2 (|x − x H |κ + |x H − x L |κ + |y − x L |κ ) 1

≤ C E 2 |x − y|κ , where we used (5.17) and (5.12). The convergence of the sequence ϕ j (up to subsequences) and (iii) are now simple consequences of (5.13) (details are left to the reader). 5.3 Normal approximation The main feature of the center manifold M lies actually in the fact that it allows to make a good approximation of the current which turns out to be almost centered by M. We introduce the following deVnition. DeHnition 5.3 (M-normal approximation). An M-normal approximation of T is given by a pair (K, F) such that (A1) F : M → A Q (U) is Lipschitz and takes the special form  F(x) = [[x + Ni (x)]] , i

with Ni (x) ⊥ Tx M for every x ∈ M and i =  1, . . . , Q. (A2) K ⊂ M is closed, contains   ∩ [− 72 , 72 ]m and

The map N =

 i

TF

p−1 (K) = T

p−1 (K).

[[Ni ]] : M → A Q (U) is called the normal part of F.

As proven in [14, Theorem 2.4], the center manifold M of the previous section allows to construct an M-normal approximation which does approximate the area minimizing current T . In order to state the result, to each L ∈ W we associate a Whitney region L on M as follows:  3   7 7 m L :=  H ∩ − , , 2 2 &(L). We will use where H is the cube concentric to L with &(H ) = 17 16 N |L 0 to denote the quantity supx∈L G(N (x), Q [[0]]).

157 Higher codimension integral currents

Theorem 5.4. Let γ2 := γ41 , with γ1 the constant of Theorem 3.9. Under the hypotheses of Theorem 5.2, if ε2 is sufFciently small, then there exist constants β2 , δ2 > 0 and an M-normal approximation (K, F) such that the following estimates hold on every Whitney region L: 1

Lip(N |L ) ≤ C E γ2 &(L)γ2 and N |L C 0 ≤ C E 2m &(L)1+β2 , (5.18)  |D N |2 ≤ C E &(L)m+2−2δ2 , (5.19) L

|L \ K| + T F − T (p−1 (L)) ≤ C E 1+γ2 &(L)m+2+γ2 .

(5.20)

Moreover, for any a > 0 and any Borel set V ⊂ L, we have 

β2 γ2 |η ◦ N | ≤ C E &(L)3+ 3 + a &(L)2+ 2 |V| V   2+γ2 C G N , Q [[η ◦ N ]] . (5.21) + a V Let us brieWy explain the conclusions of the theorem. The estimates in (5.18) and (5.19) concern the regularity properties of the normal approximation N , and will play an important role in many of the subsequent arguments. However, the key properties of N are (5.20) and (5.21): the former estimates the error done in the approximation on every Whitney region; while the latter estimates the L 1 norm of the average of N , which is a measure of the centering of the center manifold. Note that both estimates are in some sense “superlinear” with respect to the relevant parameters: indeed, as it will be better understood later on, they involve either a superlinear power of the excess E 1+γ2 or the L 2+γ2 norm of N (which is of higher order with respect to the “natural” L 2 norm). 5.4 Construction criteria The estimates and the results of the previous two subsections depend very much on the way the Whitney decomposition, the interpolating functions and the normal approximation are constructed. We start recalling the notation p L = (x L , yL ) where L is a dyadic cube, x L its center and yL ∈ π0⊥ is chosen in such a way that p L ∈ spt (T ). Moreover, we set B L := B64r L ( p L ), √ where we recall that r L := M0 m &(L) for some large constant M0 ∈ N. We deVne the families of cubes of the Whitney decomposition W = We ∪ Wh ∪ Wn

and S ⊂ C .

158 Emanuele Spadaro

We use the notation S j = S ∩ C j , W j = W ∩ C j and so on. We recall the notation for the excess, E(T, Br (x)) := min E(T, Br (x), τ ) = E(T, Br (x), π). τ

The m-dimensional planes π realizing the minimum above are called optimal planes of T in a ball Br (x) if, in addition, π optimizes the height among all planes that optimize the excess:   h(T, Br (x), π) = min h(T, Br (x), τ ) : τ satisVes (5.3) (5.22) =: h(T, Br (x)). An optimal plane in the ball B L is denoted by π L . We Vx a big natural number N0 , and constants Ce , Ch > 0, and we deVne W i = S i = ∅ for i < N0 . We proceed with j ≥ N0 inductively: if the father of L ∈ C j is not in W j−1 , then (EX) L ∈ W ej if E(T, B L ) > Ce E &(L)2−2δ2 ; 1 j j (HT) L ∈ W h if L ∈ We and h(T, B L ) > Ch E 2m &(L)1+β2 ; j (NN) L ∈ W nj if L ∈ W ej ∪ W h but it intersects an element of W j−1 ; if none of the above occurs, then L ∈ S j . We Vnally set  := [−4, 4]m \

L= L∈W

"

L.

(5.23)

j≥N0 L∈S j

Observe that, if j > N0 and L ∈ S j ∪ W j , then necessarily its father belongs to S j−1 . For what concerns the interpolating functions g L , they are obtained as the result of the following procedure. (1) Let L ∈ S ∪W and π L be an optimal plane. Then, T C¯ 32r L ( p L ,π L ) fulVlls the assumptions of the approximation Theorem 3.9 in the cylinder C¯ 32r L ( p L , π L ), and we can then construct a Lipschitz approximation f L : B8r L ( p L , π L ) → A Q (π L⊥ ). (2) We let h L : B7r L ( p L , π L ) → π L⊥ be a regularization of the average given by h L := (η ◦ f L ) ∗ &(L) ,   where  ∈ Cc∞ (B1 ) is radial,  = 1 and |x|2 (x) dx = 0.

159 Higher codimension integral currents

(3) Finally, we Vnd a smooth map g L : B4r L ( p L , π0 ) → π0⊥ such that G g L = Gh L

C¯ 4r L ( p L , π0 ),

where we recall that Gu denotes the current induced by the graph of a function u. The fact that the above procedure can be applied follows from the choice of the stopping criteria for the construction of the Whitney decomposition. We refer to [14] for a detailed proof. Here we only stress the fact that this construction depends strongly on the choice of the constants involved: in particular, Ce , Ch , β2 , δ2 , M0 are positive real numbers and N0 a natural number satisfying in particular   1 γ1 β2 = 4 δ2 = min , , (5.24) 2m 100 where γ1 is the constant of Theorem 3.9, and √ ¯ Q) ≥ 4 and m M0 27−N0 ≤ 1 . M0 ≥ C0 (m, n, n,

(5.25)

Note that β2 and δ2 are Vxed, while the other parameters are not Vxed but are subject to further restrictions in the various statements, respecting a very precise “hierarchy” (cp. [14, Assumption 1.9]). Finally, we add also a few words concerning the construction of the normal approximation N . In every Whitney region L the map N is a suitable extension of the reparametrization of the Lipschitz approximation f L . Then the estimates (5.18), (5.19) and (5.20) follow easily from Theorem 3.9. The most intricate proof is the one of (5.21) for which the choice of the regularization h L deeply plays a role. The main idea is that, on the optimal plane π L , the average of the sheets of the minimizing current is almost the graph of a harmonic function. Therefore, a good way to regularize it (which actually would keep it unchanged if it were exactly harmonic) is to convolve with a radial symmetric molliVer. This procedure, which we stress is not the only possible one, will indeed preserve the main properties of the average. 5.5 Splitting before tilting The above criteria are not just important for the construction purposes, but also lead to a couple of important estimates which will be referred to as splitting-before-tilting estimates. Indeed, it is not a case that the powers of the side-length in the (EX) and (HT) criteria look like the powers in the familiar decay of the excess and in the height bound. In fact it turns

160 Emanuele Spadaro

out that, following the arguments for the height bound and for the decay of the excess, one can infer two further consequences of the Whitney decomposition’s criteria. If a dyadic cube L has been selected by the Whitney decomposition procedure for the height criterion, then the M-normal approximation above the corresponding Whitney region needs to have a large pointwise separation (see (5.28) below).

5.5.1 (HT)-cubes

Proposition 5.5 ((HT)-estimate). If ε2 is sufFciently small, then the following conclusions hold for every L ∈ W h : !(T, p) ≤ Q −

1 2

∀ p ∈ B16r L ( p L ),

1 L ∩ H = ∅ ∀ H ∈ W n with &(H ) ≤ &(L); 2   1 1 1+β G N (x), Q [[η ◦ N (x)]] ≥ Ch E 2m &(L) 2 ∀ x ∈ L. 4 A simple corollary of the previous proposition is the following.

(5.26) (5.27) (5.28)

Corollary 5.6. Given any H ∈ W n there is a chain L = L 0 , L 1 , . . . , L j = H such that: (a) L 0 ∈ W e and L i ∈ W n for all i = 1, . . . , j; (b) L i ∩ L i−1  = ∅ and &(L i ) = 12 &(L i−1 ) for all i = 1, . . . , j. In particular, H ⊂ B3√m&(L) (x L , π0 ). We use this last corollary to partition W n . DeHnition 5.7 (Domains of inIuence). We Vrst Vx an ordering of the cubes in W e as {Ji }i∈N so that their side-length decreases. Then H ∈ W n belongs to W n (J0 ) if there is a chain as in Corollary 5.6 with L 0 = J0 . Inductively, W n (Jr ) is the set of cubes H ∈ W n \ ∪i 0 is a suitably chosen constant, always assumed to be smaller than ε2 . Observe that, if (sk ) ⊂ R and sk ↑ s, then s ∈ R. We cover R with a collection F = {I j } j of intervals I j =]s j , t j ] deVned as follows: we start with t0 := max{t : t ∈ R}. Next assume, by induction, to have deVned t0 > s0 ≥ t1 > s1 ≥ . . . > s j−1 ≥ t j , and consider the following objects: - T j := ((ι0,t j )" T ) B6√m , and assume (without loss of generality, up to a rotation) that E(T j , B6√m , π0 ) = E(T j , B6√m ); - let M j the corresponding center manifold for T j , given as the graph of a map ϕ j : π0 ⊃ [−4, 4]m → π0⊥ , (for later purposes we set  j (x) := (x, ϕ j (x))).

162 Emanuele Spadaro

Then, one of the following possibilities occurs: (Stop) either there is r ∈]0, 3] and a cube L of the Whitney decomposition W ( j) of [−4, 4]m ⊂ π0 (applied to T j ) such that &(L) ≥ cs r

and

L ∩ B¯ r (0, π0 )  = ∅;

(5.32)

(Go) or there exists no radius as in (Stop). It is possible to show that when (Stop) occurs for some r, such r is smaller than 2−5 . This justiVes the following: (1) in case (Go) holds, we set s j := 0, i.e. I j :=]0, t j ], and end the procedure; (2) in case (Stop) holds we let s j := r¯ t j , where r¯ is the maximum radius satisfying (Stop). We choose then t j+1 as the largest element in R∩]0, s j ] and proceed iteratively. s

The following are easy consequences of the deVnition: for all r ∈] t jj , 3[, it holds E(T j , Br ) ≤ Cε32 r 2−2δ2 ,

(5.33) 1 2m

j 1+β2 , (5.34) sup{dist(x, M j ) : x ∈ spt (T j ) ∩ p−1 j (Br ( p j ))} ≤ C (E ) r

where E j := E(T j , B6√m ) and p j denotes the nearest point projection on M j deVned on a neighborhood of the center manifold (for the proof we refer to [15]). 5.7 Families of subregions Let M be a center manifold and  : π0 → Rm+n the paremetrizing map. Set q := (0) and denote by B the projection of the geodesic ball 1/m pπ0 (Br (q)), for some r ∈ (0, 4). Since ϕC 3,κ ≤ Cε2 in Theorem 5.2, it is simple to show that B is a C 2 convex set and that the maximal curvature of ∂ B is everywhere smaller than r2 . Thus, for every z ∈ ∂ B there is a ball Br/2 (y) ⊂ B whose closure touches ∂ B at z. In this section we show how one can partition the cubes of the Whitney decomposition which intersect B into disjoint families which are labeled by pairs (L , B(L)) cube–ball enjoying different properties.

163 Higher codimension integral currents

Proposition 5.9. There exists a set Z of pairs (L , B(L)) with this properties: , (i) if (L , B(L)) ∈ Z, then L ∈ We ∪ Wh , the radius of B(L) is &(L) 4 ; B(L) ⊂ B and dist (B(L), ∂ B) ≥ &(L) 4 (ii) if the pairs (L , B(L)), (L  , B(L  )) ∈ Z are distinct, then L and L  are distinct and B(L) ∩ B(L  ) = ∅; (iii) the cubes W which intersect B are partitioned into disjoint families W(L) labeled by (L , B(L)) ∈ Z such that, if H ∈ W(L), then H ⊂ B30√m&(L) (x L ). In this way, every cube of the Whitney decomposition intersecting B can be uniquely associated to a ball B(L) ⊂ B for some L ∈ W e ∩ W h . This will allow to transfer the estimates form the cubes of the Whitney decomposition to the ball B. 5.7.1 Proof of Proposition 5.9

We start deVning appropriate families of

cubes and balls. DeHnition 5.10 (Family of cubes). We Vrst deVne a family T of cubes in the Whitney decomposition W as follows: (i) T includes all L ∈ Wh ∪ We which intersect B; (ii) if L  ∈ Wn intersects B and belongs to the domain of inWuence Wn (L) of the cube L ∈ We as in DeVnition 5.7, then L ∈ T . It is easy to see that, if r belongs to an interval of Wattening,√then for every L ∈ T it holds that &(L) ≤ 3cs r ≤ r and dist(L , B) ≤ 3 m &(L). Therefore, we can also deVne the following associated balls. DeHnition 5.11. For every L ∈ T , let x L be the center of L and: (a) if x L ∈ B, we then set s(L) := &(L) and B L := Bs(L) (x L , π); (b) otherwise we consider the ball Br(L) (x L , π) ⊂ π such that its closure touches B at exactly one point p(L), we set s(L) := r(L) + &(L) and deVne B L := Bs(L) (x L , π). We proceed to select a countable family T of pairwise disjoint balls {B L }. We let S := sup L∈T s(L) and start selecting a maximal subcollection T1 of pairwise disjoint balls with radii larger than S/2. Clearly, T1 is Vnite. In general, at the stage k, we select a maximal subcollection Tk of pairwise disjoint balls which do not intersect any of the previously . . . ∪ Tk−1 and which have radii r ∈]2−k S, 21−k S]. selected balls in T1 ∪Finally, we set T := k Tk .

164 Emanuele Spadaro

DeHnition 5.12 (Family of pairs cube-balls (L ,B(L)) ∈ Z ). Recalling the convexity properties of B and &(L) ≤ r, it easy to see that there exist balls B&(L)/4 (q L , π) ⊂ B L ∩ B which lie at distance at least &(L)/4 from ∂ B. We denote by B(L) one of such balls and by Z the collection of pairs (L , B(L)) with B L ∈ T. Next, we partition the cubes of W which intersect B into disjoint families W(L) labeled by (L , B(L)) ∈ Z in the following way. Let H ∈ W have nonempty intersection with B. Then, either H is in T and we set J := H , or is in the domain of inWuence of some√J ∈ T . If J  = H , then the separation between J and H is at most 3 m&(J ) and, hence, H ⊂ B4√m&(J ) (x J ). By construction there is a B L ∈ T with B J ∩ B L  = ∅ and radius s(L) ≥ s(J2 ) . We then prescribe H ∈ W(L). Observe that √ s(L) ≤ 4 m &(L) and s(J ) ≥ &(J ). Therefore, it also holds √ √ &(J ) ≤ 8 m &(L) and |x J − x L | ≤ 5s(L) ≤ 20 m &(L), thus implying H ⊂ B4√m &(J ) (x J ) ⊂ B4√m&(J )+20√m &(L) (x L ) ⊂ B30√m &(L) (x L ) .

6 Order of contact In this section we discuss the issues in steps (D) and (E) of the sketch of proof in Section 2.7, i.e. the order of contact of the normal approximation with the center manifold. The key word for this part is frequency function, which is the monotone quantity discovered by Almgren controlling the vanishing order of a harmonic function. In order to explain this point, we consider Vrst the case of a real valued harmonic function f : B1 ⊂ R2 → R with an expansion in polar coordinates f (r, θ) = a0 +

∞ 

  r k ak cos(kθ) + bk sin(kθ) .

k=1

How can one detect the smallest index k such that ak or bk is not 0? It is not difVcult to show that the quantity  r Br |∇ f |2 (6.1) I f (r) :=  2 ∂ Br | f |

165 Higher codimension integral currents

is monotone increasing in r and its limit as r ↓ 0 gives exactly the smallest non-zero index in the expansion above. I f is what Almgren calls the frequency function (and the reason for such terminology is now apparent from the example above), and one of the most striking discoveries of Almgren is that the monotonicity of the frequency remains true for Q-valued functions and in fact allows to obtain a non-trivial blowup limit. In the next subsections, we see how this discussion generalizes to the case of area minimizing currents, where an almost monotonicity formula can be derived for a suitable frequency deVned for the M-normal approximation. 6.1 Frequency function’s estimate For every interval of Wattening I j =]s j , t j ], let N j be the normal approximation of T j on M j . Since the L 2 norm of the trace of N j may not have any connection to the current itself (remember that N j misses a set of positive measure of T j ), we need to introduce an averaged version of the frequency function. To this aim, consider the following piecewise linear function ϕ : [0 + ∞[→ [0, 1] given by ⎧ ⎪ ⎨1 ϕ(r) := 2 − 2r ⎪ ⎩ 0

for r ∈ [0, 12 ], for r ∈ ] 12 , 1], for r ∈ ]1, +∞[,

and let us deVne a new frequency function in the following way. DeHnition 6.1. For every r ∈]0, 3] we deVne: 

 D j (r) := and

Mj

d j ( p) ϕ r

 H j (r) := −

Mj

ϕ







d j ( p) r

|D N j |2 ( p) dp, 

|N j |2 ( p) dp , d( p)

where d j ( p) is the geodesic distance on M j between p and  j (0). If we have that H j (r) > 0, then we deVne the frequency function I j (r) :=

r D j (r) . H j (r)

166 Emanuele Spadaro

Note that, by the Coarea formula,  H j (r) = 2  =2

Br \Br/2 ( j (0)) r

1 t

r/2



|N |2 d( p)

∂ Bt ( j (0))

|N j |2 dt ,

(6.2)

whereas, using Fubini,  r D j (r) =

 |D N j | (x)

r

2

Mj

=2

 r Bt ( j (0))

r 2

r 2

1]|x|,∞[ (t) dt dHm (x)

|D N j |2 dt.

(6.3)

This explains in which sense I j is an average of the quantity introduced by F. Almgren. The main analytical estimate is then the following. Theorem 6.2. If ε3 in (5.31) is sufFciently small, then there exists a constant C > 0 (indepent of j) such that, if [a, b] ⊂ [ st , 3] and H j |[a,b] > 0, then it holds (6.4) I j (a) ≤ C(1 + I j (b)). To simplify the notation, we drop the index j and omit the measure Hm in the integrals over regions of M. For the proof of the theorem we need to introduce some auxiliary functions (all absolutely continuous with respect to r). We let ∂rˆ denote the derivative along geodesics starting at (0). We set  E(r) := −

M

ϕ



d( p) r





Q

 Ni ( p), ∂rˆ Ni ( p) dp , i=1

ϕ  d(rp) d( p) |∂rˆ N ( p)|2 dp, M

(r) := ϕ d(rp) |N |2 ( p) dp .

G(r) := −

M

The proof of Theorem 6.2 exploits some “integration by parts” formulas, which in our setting are given by the Vrst variations for the minimizing current. We collect these identities in the following proposition, and proceed then with the proof of the theorem.

167 Higher codimension integral currents

Proposition 6.3. There exist dimensional constants C, γ3 > 0 such that, if the hypotheses of Theorem 6.2 hold and I ≥ 1, then    H (r) − m−1 H(r) − 2 E(r) ≤ CH(r), (6.5) r r   D(r) − r −1 E(r) ≤ CD(r)1+γ3 + Cε2 (r), (6.6) 3    D (r) − m−2 D(r) − 22 G(r) ≤ CD(r) + CD(r)γ3 D (r) r r +r −1 D(r)1+γ3 , 

(r) + r  (r) ≤ C r D(r) ≤ 2

(6.7) Cr 2+m ε32 .

(6.8)

We assume for the moment the proposition and prove the theorem. Proof of Theorem 6.2. It enough to consider the case in which I > 1 on ]a, b[. Set (r) := log I(r). By Proposition 6.3, if ε3 is sufVciently small, then D(r) E(r) ≤ ≤ 2 D(r), (6.9) 2 r from which we conclude that E > 0 over the interval ]a, b [. Set for simplicity F(r) := D(r)−1 − rE(r)−1 , and compute − (r) =

H (r) D (r) 1 (6.6) H (r) rD (r) 1 − − = − − D (r)F(r) − . H(r) D(r) r H(r) E(r) r

Again by Proposition 6.3: H (r) (6.5) m − 1 2 E(r) ≤ +C + , H(r) r r H(r) (6.6)

|F(r)| ≤ C

(6.10)

r(D(r)1+γ3 + (r)) (6.9) (r) ≤ C D(r)γ3 −1 + C , (6.11) D(r) E(r) D(r)2

  rD (r) (6.7) m − 2 rD(r) 2 G(r) − ≤ C− − E(r) r E(r) r E(r) γ3  1+γ3 rD(r) D (r) + D(r) +C E(r) m−2 C 2 G(r) ≤C− + D(r)|F(r)| − r r r E(r) D(r)γ3 + CD(r)γ3 −1 D (r) + C r (6.8), (6.11) m − 2 2 G(r) ≤ C− − + CD(r)γ3 −1 D (r) + C r γ3 m−1 , r r E(r) (6.12)

168 Emanuele Spadaro

where we used the rough estimate D(r) ≤ C r m+2−2δ2 coming from (5.19) of Theorem 5.4 and the condition (Stop). By Cauchy-Schwartz, we have E(r) G(r) ≤ . rH(r) rE(r)

(6.13)

Thus, by (6.10), (6.12) and (6.13), we conclude − (r) ≤ C + C r γ3 m−1 + CrD(r)γ3 −1 D (r) − D (r)F(r) (6.11) (r)D (r) ≤ C r γ3 m−1 + CD(r)γ3 −1 D (r) + C . D(r)2

(6.14)

Integrating (6.14) we conclude: (a) − (b) ≤ C + C (D(b)γ3 − D(a)γ3 ) 3   b  (6.8) (a) (b)  (r) +C − + dr ≤ C. D(a) D(b) a D(r)

6.1.1 Proof of Proposition 6.3 The remaining part of this subsection is devoted to give some arguments for the proof of the Vrst variation formulas. The estimate (6.5) follows from a straightforward computation: using the area formula and setting y = r z, we have  ϕ  (|z|) m−1 |N |2 (exp(r z)) J exp(r z) dx, H(r) = −r |z| Tq M

and differentiating under the integral sign, we easily get (6.5):  ϕ  (|z|)  m−2 H (r) = − (m − 1) r |N |2 (exp(r z)) J exp(r z) dz |z| Tq M   Ni , ∂rˆ Ni  (exp(r z)) J exp(r z) dz − 2 r m−1 ϕ  (|z|) 

Tq M

i 

ϕ (|z|) d |N |2 (exp(r z)) J exp(r z) dz |z| dr Tq M m−1 2 = H(r) + E(r) + O(1) H(r), r r − r m−1

where we the following simple fact for the Jacobian of the exponential map drd J exp(r z) = O(1), because M is a C 3,κ submanifold and the exponential map exp is a C 2,κ map.

169 Higher codimension integral currents

Similarly, (6.8) follows by simple computation which involve a Poincar´e inequality: namely, if I ≥ 1, then  |N |2 ≤ C r 2 D(r). (6.15) Br (q)

We refer to [15] for the details of the proof. Here we try to explain the remaining two estimates, which instead are connected to the Vrst variation δT (X) of the area minimizing current T along a vector Veld X. The idea is the following: since the Vrst variations of T are zero, we compute them using its approximation N and derive the integral equality in the Proposition 6.3. To understand the meaning of these estimates, consider u : Rm → Rn a harmonic function. Then, computing the variations of the Dirichlet energy of u leads to the following two identities:   ∂u 2 |Du| = u· , ∂ν Br ∂ Br   2    ∂u  m−2 2 2   , |Du| = |Du| + 2  ∂ν  r ∂ Br

Br

∂ Br

which are the exact analog of (6.6) and (6.7) without any error term. What we need to do is then to replace the Dirichlet energy with the area functional, and to consider the fact that the normal approximation N is only approximately stationary with respect to this functional. We start Vxing a tubular neighborhood U of M and the normal projection p : U → M. Observe that p ∈ C 2,κ . We will consider:

(1) the outer variations, where X ( p) = X o ( p) := ϕ d(p(r p)) ( p − p( p)). (2) the inner variations, where X ( p) = X i ( p) := Y (p( p)) with   d( p) ∂ d( p) Y ( p) := ϕ ∀ p ∈ M. r r ∂ rˆ  Consider now the map F( p) := i [[ p+ Ni ( p)]] and the current T F associated to its image. Observe that X i and X o are supported in p−1 (Br (q)) but none of them is compactly supported. However, it is simple to see that δT (X) = 0. Then, we have |δT F (X)| = |δT F (X) − δT (X)|       div X  dT  + div X  dT F , (6.16) ≤ T TF spt (T )\Im(F) Im(F)\spt (T ) 67 8 5 Err4

170 Emanuele Spadaro

 where Im(F) is the image of the map F(x) = i [[(x, Ni (x))]], i.e. the support of the current T F .   Set now for simplicity ϕr ( p) := ϕ d(rp) . It is not hard to realize that the mass of the current T F can be expressed in the following way:   1 m M(T F ) = Q H (M) − Q H, η ◦ N  + |D N |2 2 M M  

+ P2 (x, Ni ) + P3 (x, Ni , D Ni ) + R4 (x, D Ni ) , M

i

(6.17) where P2 , P3 and R4 are quadratic, cubic and fourth order errors terms (see [16, Theorem 3.2]) One can then compute the Vrst variation of a push-forward current T F and obtain (cp. [16, Theorem 4.2])  δT F (X o ) =

M

Q 3

  Ni ⊗ ∇ϕr : D Ni + Erroj , (6.18) ϕr |D N |2 + i=1

j=1

where the errors Erroj satisfy  Erro1  |Erro3 | ≤ C

M

= −Q

|Erro2 | ≤ C

M

ϕr HM , η ◦ N ,

(6.19)

|ϕr ||A|2 |N |2 ,

(6.20)

M

   |N ||A| + |D N |2 |ϕr ||D N |2 + |Dϕr ||D N ||N | , (6.21)

here HM is the mean curvature vector of M. Plugging (6.18) into (6.16), we then conclude 4      Erro  , D(r) − r −1 E(r) ≤ j

(6.22)

j=1

where Erro4 corresponds to Err4 of (6.16) when X = X o . Arguing similarly with X = X i (cp. [16, Theorem 4.3]), we get 1 δT F (X i ) = 2



+

M

Q

 2 D Ni : (D Ni · DM Y ) |D N | divM Y − 2

3  j=1

i=1

Errij ,

(6.23)

171 Higher codimension integral currents

where this time the errors Errij satisfy    i Err1 = −Q HM , η ◦ N  divM Y + DY HM , η ◦ N  , M    i |Err2 | ≤ C |A|2 |DY ||N |2 + |Y ||N | |D N | , M

|Erri3 |

(6.24) (6.25)



  |Y ||A||D N |2 |N | + |D N | M  

+ |DY | |A| |N |2 |D N | + |D N |4 .

≤C

(6.26)

Straightforward computations lead to      d( p) ∂ ∂ d( p) Id  d( p) ⊗ +ϕ + O(1) , DM Y ( p) = ϕ r r 2 ∂ rˆ ∂ rˆ r r (6.27) divM Y ( p) = ϕ





d( p) r



 

d( p) m d( p) + ϕ + O(1) . (6.28) r2 r r

Plugging (6.27) and (6.28) into (6.23) and using (6.16) we then conclude 4       Erri  . (6.29) D (r) − (m − 2)r −1 D(r) − 2r −2 G(r) ≤ CD(r) + j j=1

Proposition 6.3 is then proved by the estimates of the errors terms done in the next subsection. We consider the family of pairs Z = {(Ji , B(Ji ))}i∈N introduced in the previous section, and set

6.1.2 Estimates of the errors terms

Bi := (B(Ji )) and Ui = ∪ H ∈W(Ji ) (H ) ∩ Br (q) . Set Vi := Ui \ K, where K is the coincidence set of Theorem 5.4. By a simple application of Theorem 5.4 we derive the following estimates:   γ 2+m+ 22 |η ◦ N | ≤ C E &i +C |N |2+γ2 , (6.30) Ui Ui  |D N |2 ≤ C E &im+2−2δ2 , (6.31) Ui

N C 0 (Ui ) +

sup

1

p∈spt (T )∩p−1 (Ui )

1+β2

| p − p( p)| ≤ C E 2m &i

,

γ

Lip(N |Ui ) ≤ C E γ2 &i 2 , M(T

−1

p (Vi )) + M(T F

−1

p (Vi )) ≤

m+2+γ2 C E 1+γ2 &i .

(6.32) (6.33) (6.34)

172 Emanuele Spadaro

Observe that the separation between Bi and ∂Br(q) is larger than &(Ji )/4 by Proposition 5.9 (i), and then ϕr ( p) = ϕ d(rp) satisVes inf ϕr ( p) ≥ (4r)−1 &i ,

p∈Bi

(6.35)

where &i := &(Ji ). From this and Proposition 5.9 (iii), we also obtain C (6.35) &i ≤ C inf ϕr ( p) , p∈Bi r

sup ϕr ( p) − inf ϕr ( p) ≤ CLip(ϕr )&i ≤ p∈Ui

p∈Ui

which translates into sup ϕr ( p) ≤ C inf ϕr ( p) . p∈Bi

p∈Ui

(6.36)

Moreover, by an application of the splitting-before-tilting estimates in Proposition 5.5 and Proposition 5.8, we infer that  1 m+2+2β 2 |N |2 ≥ c E m &i if L i ∈ Wh , (6.37) i B  |D N |2 ≥ c E &im+2−2δ2 if L i ∈ We . (6.38) Bi

This easily implies the following estimates under the hypotheses I ≥ 1: by applying (6.15), (6.35), (6.37) and (6.38), we get, for suitably chosen γ (t), C(t) > 0,

γ (t) 

2t t   t t 2 ϕr (|D N |2 + |N |2 ) sup E &i + inf ϕr &i ≤ C(t) sup Bi

i

Bi

i

γ (t)

≤ C(t)D(r) and similarly  i

  m+2+ γ42 inf ϕr E &i ≤C Bi

i

,

 Bi

ϕr (|D N |2 + |N |2 )

≤ CD(r) ,  i

E

m+2+ &i

γ2 4

 ≤C

Br (q)

(6.39)

(6.40)

  |D N |2 + |N |2

  ≤ C D(r) + rD (r) .

(6.41)

We can now pass to estimate the errors terms in (6.6) and (6.7) in order to conclude the proof of Proposition 6.3.

173 Higher codimension integral currents

Errors of type 1. By Theorem 5.2, the map ϕ deVning the center man1 ifold satisVes DϕC 2,κ ≤ C E 2 , which in turn implies HM  L ∞ + 1 D HM  L ∞ ≤ C E 2 (recall that HM denotes the mean curvature of M). Therefore, by (6.36), (6.30), (6.40) and (6.39), we get  o Err  ≤ C 1

 M 1

≤ C E2

ϕr |HM | |η ◦ N |  

 2+m+γ2 +C sup ϕr E & j



Ui

j



≤ CD(r)1+γ3 + C

1

γ (1+β2 )

ϕr |N |2+γ2

Uj





E 2 & j2

Uj

j

ϕr |N |2 ≤ CD(r)1+γ3 ,

and analogously  i Err  ≤ C r −1 1 ≤Cr

−1



  |HM | + |DY HM | |η ◦ N |

M

E

1 2



E

2+m+γ2 &j

 +C

j

Uj

|N |2+γ2



  ≤ C r −1 D(r)γ D(r) + r D (r) . 1

Errors of type 2. From AC 0 ≤ CDϕC 2 ≤ C E 2 ≤ Cε3 , it follows that Erro2 ≤ Cε32 (r). Moreover, since |D X i | ≤ Cr −1 , (6.15) leads to  i Err  ≤ Cr −1 2



 Br ( p0 )

|N |2 + C

ϕr |N ||D N | ≤ CD(r) .

Errors of type 3. Clearly, we have  o Err  ≤ 3



  ϕr |D N |2 |N | + |D N |4 +C r −1 5 67 8 5 + C r −1 5



I1

 |D N |3 |N | Br (q) 67 8 I2

|D N ||N |2 . 67 8

Br (q)

I3

174 Emanuele Spadaro

We estimate separately the three terms (recall that γ2 > 4δ2 ):  γ  m+2+ 22 I1 ≤ ϕr (|N |2 |D N | + |D N |3 ) ≤ I3 + C sup ϕr E 1+γ2 & j Br ( p0 )

(6.40) & (6.39)

≤

Uj

j

I3 + CD(r)1+γ3 ,

I2 ≤ Cr −1



m+3+β2 +

1

E 1+ 2m +γ2 & j

γ2 2

j

(6.36)

≤ C



1

m+2+β2 +

E 1+ 2m +γ2 & j

I3 ≤ Cr



γ E γ2 & j 2

j

inf ϕr

(6.40) & (6.39)

≤

Bj

j

−1

γ2 2

 |N |

2

Uj

(6.39)

≤ Cr

−1

γ3

D(r)

CD(r)1+γ3 ,

 Br (q)

|N |2

(6.15)

≤ CD(r)1+γ3

For what concerns the inner variations, we have   −1  i |Err3 | ≤ C r |D N |3 + r −1 |D N |2 |N | + r −1 |D N ||N |2 . Br (q)

The last integrand corresponds to I3 , while the remaining part can be estimated as follows:    1 1+β 3 2 γ2 γ2 −1 −1 2 r (|D N | +|D N | |N |) ≤ C r (E & j + E 2m & j ) |D N |2 Br (q)

j (6.39)



Uj

≤ C r −1 D(r)γ3 |D N |2 Br (q)   ≤ CD(r)γ3 D (r) + r −1 D(r) .

Errors of type 4. We compute explicitly |Dd(p( p), q)| |D X o ( p)| ≤ 2 | p − p( p)| + ϕr ( p) |D( p − p( p))| r   | p − p( p)| ≤C + ϕr ( p) . r

175 Higher codimension integral currents

It follows readily from (6.16), (6.32) and (6.34) that

 1 m+2+γ2 1+β |Erro4 | ≤ C r −1 E 2m &i 2 + sup ϕr E 1+γ2 &i i (6.35) & (6.36)

≤

C

3

γ2 4

E γ2 &i

i (6.40) &(6.39)

≤

Ui



m+2+

inf ϕr E &i

γ2 4

Bi

CD(r)1+γ3 .

(6.42)

Similarly, since |D X i | ≤ Cr −1 , we get Erri4 ≤ Cr −1



γ2

γ m+2+ 22 E γ2 & j2 E & j

j (6.41) & (6.39)

≤

  CD(r)γ D (r) + r −1 D(r) .

Remark 6.4. Note that the “superlinear” character of the estimates in Theorem 5.4 has played a fundamental role in the control of the errors. 6.2 Boundness of the frequency We have proven in the previous subsection that the frequency of the Mnormal approximation remains bounded within a center manifold in the corresponding interval of Wattening. In order to pass into the limit along the different center manifolds, we need also to show that the frequency remains bounded in passing from one to the other. This is again a consequence of the splitting-before-tilting estimates and we provide here some details of the proof, referring to [14] for the complete argument. To simplify the notation, we set p j :=  j (0) and write simply Bρ in place of Bρ ( p j ) . Theorem 6.5 (Boundedness of the frequency functions). If the intervals of Gattening are inFnitely many, then there is a number j0 ∈ N such that s

H j > 0 on ] t jj , 3[ for all j ≥ j0

and

sup sup I j (r) < ∞ . (6.43) j≥ j0 r∈] s j ,3[ tj

Sketch of the proof. We partition the extrema t j of the intervals of Wattening into two different classes: (A) those such that t j = s j−1 ; (B) those such that t j < s j−1 .

176 Emanuele Spadaro

Let L ∈ W ( j−1) be a cube of the Whitney decomposition such that cs r ≤ &(L) and L ∩ B¯ r (0, π) = ∅. Since this cube of the Whitney decomposition at step j − 1 has size comparable with the distance to the origin, and the next center manifold starts at a comparable radius, the splitting property of the normal approximation needs to hold also for the new approximation: namely, one can show that there exists a constant c¯s > 0 such that  |N j |2 ≥ c¯s E j := E(T j , B6√m ),

If t j belongs to (A), set r :=

s j−1 . t j−1

B2 ∩M j

which obviously gives H N j (3) ≥ cE j , and than I N j (3) is smaller than a given constant, independent of j, thus proving the theorem. In the case t j belongs to the class (B), then, by construction there is η j ∈]0, 1[ such that E((ι0,t j )" T, B6√m(1+η j ) ) > ε32 . Up to extracting a subsequence, we can assume that (ι0,t j )" T converges to a cone S: the convergence is strong enough to conclude that the excess of the cone is the limit of the excesses of the sequence. Moreover (since S is a cone), the excess E(S, Br ) is independent of r. We then conclude ε32 ≤ lim inf E(T j , B3 ) . j→∞, j∈(B)

Thus, it follows again from the splitting phenomenon (see for details [15, Lemma 5.2]) that lim inf j→∞, j∈(B) H N j (3) > 0. Since D N j (3) ≤ C E j ≤ Cε32 , we achieve that lim sup j→∞, j∈(B) I N j (3) > 0, and conclude as before.

7 Final blowup argument We are now ready for the conclusion of the blowup argument, i.e. for the discussion of steps (F) and (G) of Section 2.7. To this aim we recall here the main results obtained so far. We start with an m-dimensional area minimizing integer rectiVable T in Rm+n with ∂ T = 0 and 0 ∈ D Q (T ), such that there exists a sequence of radii rk ↓ 0 satisfying lim E(T0,rk , B10 ) = 0, k→+∞ m−2+α H∞ (D Q (T0,rk ) ∩ B1 ) >

lim η > 0, k→+∞   Hm (B1 ∩ spt (T0,rk )) \ D Q (T0,rk ) > 0 ∀ k ∈ N,

(7.1) (7.2) (7.3)

for some constant α, η > 0. In the process of solving the centering problem for such currents we have obtained the following:

177 Higher codimension integral currents

1. the intervals of Wattening I j =]s j , t j ], 2. the center manifolds M j , 3. the M j -normal approximations N j : M j → A Q (Rm+n ), satisfying the conclusions of Theorem 5.2 and Theorem 5.4. It follows from the very deVnition of intervals of Wattening that each rk has to belong to one of these intervals. Therefore, in order to Vx the ideas and to simplify the notation, we will in the sequel assume that there are inVnitely many intervals of Wattening and that rk ∈ Ik : note that this is not a serious restriction, and everything holds true also in the case of Vnitely many intervals of Wattening. By the analysis of the order of contact and the estimate on the frequency function, see Theorem 6.2 and Theorem 6.5, we have also derived the information I (r) < +∞. (7.4) sup sup   j j∈N

r∈

sj tj

,3

The ultimate consequence of this estimate, thus clarifying the discussion about the non-triviality of the blowup process, is the following proposition. Proposition 7.1 (Reverse Sobolev). There exists  a constant  C > 0 with 3rj rj this property: for every j ∈ N, there exists θ j ∈ 2 t j , 3 t j such that 

 |D N j | ≤ C 2

Bθ j ( j (0))

tj rj

2  Bθ j ( j (0))

|N j |2 .

(7.5)

r

Proof. Set for simplicity r := t jj and drop the subscript j in the sequel. Using (6.2), (6.3) and (7.4), there exists C > 0 such that  3r  3 dt |D N |2 = r D(3r) ≤ C H(3r) 3 2 Bt ((0)) 2r  3r  1 dt |N |2 . =C 3 t r ∂ B ((0)) t 2 Therefore, there must be θ ∈ [ 32 r, 3r] satisfying   C 2 |D N | ≤ |N |2 . θ ∂ Bθ ((0)) Bθ ((0))

(7.6)

This is almost the desired estimate. In oder to replace the boundary integral with a bulk integral in the right hand side of (7.6), we argue by integrating along radii in a similar way to the case of single valued functions.

178 Emanuele Spadaro

Fix indeed any σ ∈]θ/2, θ[ and any point x ∈ ∂Bθ ((0)). Consider the geodesic line γ passing through x and (0), and let γˆ be the arc on γ having one endpoint x¯ in ∂Bσ ((0)) and one endpoint equal to x. Using [13, Proposition 2.1(b)] and the fundamental theorem of calculus, we easily conclude  |N (x)| ≤ |N (x)| ¯ + |D N ||N | . γˆ

Integrating this inequality in x and recalling that σ > s/2 we then easily conclude    2 2 |N | ≤ C |N | + C |N | |D N | . ∂ Bθ ((0))

∂ Bσ ((0))

Bθ ((0))

We further integrate in σ between s/2 and s to achieve    2  |N | + θ |N | |D N | |N |2 ≤ C θ ∂ Bθ ((0))

Bθ ((0))

θ2 ≤ 2C





Bθ ((0))

|D N |2 + C

Bθ ((0))

|N |2 .

(7.7)

Combining (7.7) with (7.6) we easily conclude (7.5). 7.1 Convergence to a Dir-minimizer We can now deVne the Vnal blowup sequence, because the Reverse Sobolev inequality proven in Proposition 7.1 gives the right radius θk for assuring compactness of the corresponding maps. To this aim set r¯k := 23 θk tk ∈ [rk , 2 rk ], and rescale the current and the maps accordingly: ¯ k := ι0,¯rk /tk Mk , T¯k := (ι0,¯rk )" T and M ¯ k → Rm+n for the rescaled M ¯ k -normal approximations and N¯ k : M given by   r ¯ p t k k . N¯ k ( p) := Nk r¯k tk ¯ k . Moreover, Note that the ball Bsk ⊂ Mk is sent into the ball B 3 ⊂ M 2 via some elementary regularity theory of area minimizing currents, one deduces that 1. E(T¯k , B 1 ) ≤ CE(T, Brk ) → 0; 2 2. T¯k locally converge (and in the Hausdorff sense for what concerns the supports) to an m-plane with multiplicity Q;

179 Higher codimension integral currents

¯ k locally converge to the Wat m-plane (without loss of generality 3. M π0 ); 4. recalling (7.2), m−2+α (D Q (T¯k ) ∩ B1 ) ≥ η > 0 , H∞

(7.8)

for some positive constant η . We can then consider the following deVnition for the blow-up maps Nkb : B3 ⊂ Rm → A Q (Rm+n ) given by ¯ Nkb (x) := h−1 k Nk (ek (x)) ,

with hk :=  N¯ k  L 2 (B 3 ) ,

(7.9)

2

¯ k denotes the exponential map at ¯k →M where ek : B3 ⊂ Rm  T p¯k M p¯ k = tk k (0)/¯rk . Proposition 7.1 implies then that there exists a constant C > 0 such that, for every k,  B3

|D Nkb |2 ≤ C.

(7.10)

2

Moreover, as a simple consequence of Theorem 5.4 (details left to the readers), we Vnd an exponent γ > 0 such that

M((T F¯k

γ Lip( N¯ k ) ≤ Chk , 2+2γ − T¯k ) (p−1 , k (B 32 )) ≤ Chk  |η ◦ N¯ k | ≤ Ch2k .

B3

(7.11) (7.12) (7.13)

2

It then follows from (7.10), Nkb  L 2 (B3/3 ) ≡ 1 and the Sobolev embedding for Q-valued functions (cp. [13, Proposition 2.11]) that up to subb : sequences (as usual not relabeled) there exists a Sobolev function N∞ b m+n 2 B 3 → A Q (R ) such that the maps Nk converge strongly in L (B 3 ) to 2 2 b . Then from (7.13) we deduce also that N∞ b ≡ 0 and Nkb  L 2 (B3/3 ) ≡ 1. η ◦ N∞

(7.14)

¯ k -normal approximations and the M ¯ k conMoreover, since the N¯ k are M m b verging to the Wat m-dimensional plane R × {0}, N∞ takes values in the space of Q-points of {0} × Rn (in place of the full Rm+n ). b is To conclude our contradiction argument, we need to prove the N∞ Dir-minimizing.

180 Emanuele Spadaro

b is Dir-minimizing 7.1.1 N∞

Apart from the necessary technicalities, the proof of this claim is very intuitive and relies on the following observab could be decreased, then one would be able to tion: if the energy of N∞ Vnd a rectiVable current with less mass then T¯k , because the rescaling of Nkb are done in terms of the L 2 norm hk whereas the errors in the normal approximation are superlinear with hk . Next we give all the details for this arguments. ¯ k )⊥ , ¯ k an orthonormal frame of (T M We can consider for every M ν1k , . . . , νnk , with the property (cf. [16, Lemma A.1]) that ¯ k ) as k ↑ ∞ for every j in C 2,κ/2 (M

ν kj → em+ j

(here e1 , . . . , em+n is the standard basis of Rm+n ). ¯ k → A Q ({0} × Rn ), Given now any Q-valued map u = i [[u i ]] : M we can consider the map    (u i (x)) j ν kj (x) , uk : x → i

where we set (u i ) j := u i (x), em+ j  and we use Einstein’s convention. Then, the differential map Duk := i [[D(uk )i ]] is given by D(uk )i = D(u i ) j ν kj + (u i ) j Dν kj . Taking into account that Dνik C 0 → 0 as k → +∞, we deduce that        2  2 2   |u| + |Du|2 . (7.15) |Duk | − |Du|  ≤ o(1)  Note that Nkb has also the form ubk for some Q-valued function u bk : ¯ k → A Q ({0} × Rn ). M b . There is nothing to We now show the Dir-minimizing property of N∞ prove if its Dirichlet energy vanishes. We can therefore assume that there exists c0 > 0 such that  2 |D N¯ k |2 . (7.16) c0 hk ≤

B3 2

We argue by contradiction and assume there is a radius t ∈ function f : B 3 → A Q ({0} × Rn ) such that

5 4

 , 32 and a

2

b | B 3 \Bt f | B 3 \Bt = N∞ 2

for some δ > 0.

2

and Dir(f, Bt ) ≤ Dir(Nb∞ , Bt ) − 2 δ,

181 Higher codimension integral currents

Using f as a model, we need to Vnd a sequence of functions vkb such that they have the same boundary data of Nkb and less energy. This can be done because of the strong convergence of the traces and the possibility to make an interpolation between two functions with close by traces. This is one of the instances where thinking to multiple valued functions as classical single valued ones may be useful. In any case, the details are given in [17, Proposition 3.5] and lead to competitor functions vkb such that, for k large enough,  B3

vkb |∂ Br = Nkb |∂ Br ,  b 2 |η ◦ vk | ≤ Chk and

2

γ

B3

Lip(vkb ) ≤ Chk ,  b 2 |Dvk | ≤ |D Nkb |2 − δ h2k ,

2

where C > 0 is a constant independent of k. Clearly, setting N˜ k = vkb e−1 k satisfy N˜ k ≡ N¯ k

 B3

|η ◦ N˜ k | ≤ Ch2k

2

γ in B 3 \ Bt , Lip( N˜ k ) ≤ Chk , 2   2 ˜ and |D Nk | ≤ |D N¯ k |2 − δh2k .

B3

B3

2

Consider Vnally the map F˜k (x) =

2

  i

 x + N˜ i (x) . The current T F˜k

coincides with T F¯k on p−1 k (B 32 \ Bt ). DeVne the function ϕk ( p) =   distM¯ k (0, pk ( p)) and consider for each s ∈ t, 32 the slices T F˜k − T¯k , ϕk , s. By (7.12) we have  3 2 2+γ M(T F˜k − T¯k , ϕk , s) ≤ Chk . t

  Thus we can Vnd for each k a radius σk ∈ t, 32 on which M(T F˜k − 2+γ T¯k , ϕk , σk ) ≤ Chk . By the isoperimetric inequality (see [17, Remark 4.3]) there is a current Sk such that ∂ Sk = T F˜k − T¯k , ϕk , σk ,

(2+γ )m/(m−1)

M(Sk ) ≤ Chk

.

Our competitor current is, then, given by −1 ¯ Z k := T¯k (p−1 k (Mk \ Bσk )) + Sk + T F˜k (pk (Bσk )).

Note that Z k has the same boundary as T¯k . On the other hand, by (7.12) and the bound on M(Sk ), we have M(T˜k ) − M(T¯k ) ≤ M(T F¯k ) − M(T F˜k ) + Chk

2+2γ

.

(7.17)

182 Emanuele Spadaro

Denote by Ak and by Hk respectively the second fundamental forms and ¯ k . Using the Taylor expansion of mean curvatures of the manifolds M [16, Theorem 3.2], we achieve 

1 ¯ ˜ |D N˜ k |2 − |D N¯ k |2 M(Tk ) − M(Tk ) ≤ 2 Bρ 

+ CHk C 0 |η ◦ N¯ k | + |η ◦ N˜ k | 

2 | N¯ k |2 + | N˜ k |2 + o(h2k ) + Ak C 0 δ ≤ − h2k + o(h2k ) . 2

(7.18)

Clearly, (7.18) and (7.17) contradict the minimizing property of T¯k for k large enough and this concludes the proof. 7.2 Persistence of singularities We discuss step (G) of Section 2.7: we show that the assumptions (7.2) and (7.3) contradict Theorem 3.2, which asserts that the singular set of b has Hm−2+α measure zero. N∞ Set

  b (x) = Q [[0]] , ϒ := x ∈ B¯ 1 : N∞

b b ≡ 0 and N∞  L 2 (B 3 ) = 1, from Theorem 3.2 and note that, since η ◦ N∞ 2

m−2+α (ϒ) = 0. it follows that H∞ The main line of the contradiction argument can be summarized in three steps.

Tk ) such that 1. By (7.2) and (7.3) there exists a set k ⊂ DirQ (N m−2+α ( k ) > c2 > 0, dist( k , ϒ) > c1 > 0 and H∞

for suitable constants c1 , c2 > 0. The key aspect of the set k is the following: by the H¨older regularity of Dir-minimizing functions in Theorem 3.2, the normal approximation N¯ k must be big in modulus around any point in k . 2. Moreover, it follows from the Lipschitz approximation Theorem 3.9 (see Theorem 7.2 below that around any multiplicity Q point of the current the energy of the Lipschitz approximation is large enough with respect to the L 2 norm (cp. [17, Theorem 1.7]). This is what we call persistence of Q-point phenomenon, and is in fact the analytic core of this part of the proof.

183 Higher codimension integral currents

We moreover stress that this part of the proof (even if it is not apparent from our exposition) also uses the splitting-before-tilting estimates. 3. Putting together the previous two steps, we then conclude that there is a big part of the current where the energy of the Lipschitz approximation is large enough: matching the constant in the previous estimates, one realizes that this cannot happen on a set of positive Hm−2+α measure. As usual, the actual proof is much more involved of the heuristic scheme above. In the following we try to give some more explanations, referring to [14, 15, 17] for the detailed proof. Step (1). We cover ϒ by balls {Bσi (xi )} in such a way that 

ωm−2+α (4σi )m−2+α ≤

i

η , 2

where η > 0 is the constant in (7.8). By the compactness of ϒ, such a covering can be chosen Vnite. Let σ > 0 be a radius whose speciVc choice will be given only at the very end, and such that 0 < 40 σ ≤ min σi . Denote by k the set of Q points of T¯k far away from the singular set ϒ:   k := p ∈ D Q (T¯k ) ∩ B1 : dist( p, ϒ) > 4 min σi . 

m−2+α Clearly, H∞ ( k ) ≥ η2 . Let V denote the neighborhood of ϒ of size 2 min σi . By the H¨older continuity of Dir-minimizing functions in b (x)|2 ≥ Theorem 3.2 (ii), there is a positive constant ϑ > 0 such that |N∞ 2ϑ for every x  ∈ V. It then follows that  b 2 2ϑ ≤ − |N∞ | ∀ x ∈ B 5 with dist(x, ϒ) ≥ 3 min σi , 4

B2σ (x)

and therefore, for sufVciently large k’s,    2 G( N¯ k , Q η ◦ N¯ k )2 , ϑ hk ≤ − B2σ (x)

(7.19)

for all x ∈ k := pM¯ k ( k ). This is the claimed lower bound on the modulus of N¯ k . Step (2). This is the most important step of the proof. We start introducing the following notation. For every p ∈ k , consider z¯ k ( p) = ¯ k , where  ¯k ∈ M ¯ k is the induced parametrizapπ0 ( p) and x¯k ( p) :=  tion.

184 Emanuele Spadaro

The key claim is the following: there exists a geometric constant c0 > 0 (in particular, independent of σ ) such that, when k is large enough, for each p ∈ k there is a radius  p ≤ 2σ with the following properties:  c0 ϑ 2 1 h ≤ |D N¯ k |2 , (7.20) σ α k m−2+α B p (x¯k ( p)) p B p (x¯k ( p)) ⊂ B4 p ( p) .

(7.21)

We show here the main heuristics leading to (7.20) (and we warn the reader that these are not the complete arguments), referring to [15] for (7.21). The key estimate in this regard is the following: there exists a constant s¯ < 1 such that    2  ϑ G(N j (k) , Q η ◦ N j (k) ) ≤ |D N j (k) |2 , − m−2 4ω &(L ) m k Bs¯&(L k ) (xk ) B&(L k ) (xk ) ¯ k , there exists t ( p) ≤ &¯k such that that is, rescaling to M    2  ϑ ¯ ¯ − G( Nk , Q η ◦ Nk ) ≤ |D N¯ k |2 . 4ωm t ( p)m−2 Bt ( p) (x¯k ( p)) Bs¯t ( p) (x¯k )( p) (7.22) We show that we can choose  p ∈]¯s t ( p), 2σ [ such that (7.20) follows from (7.22). To this aim we can distinguish two cases. Either  1 (7.23) |D Nk |2 ≥ h2k , m−2 ωm t ( p) Bt ( p) (x¯k ( p)) and (7.20) follows with  p = t ( p). Or (7.23) does not hold, and we argue as follows. We use Vrst (7.22) to get    ϑ − G( N¯ k , Q η ◦ N¯ k )2 ≤ h2k . (7.24) 4 Bs¯t ( p) (x¯k ( p)) Then, we show by contradiction that there exists a radius  y ∈ [¯s t ( p), 2σ ] such that (7.20) holds. Indeed, if this were not the case, setting for sim plicity f := G( N¯ k , Q η ◦ N¯ k ) and letting j be the smallest integer such that 2− j σ ≤ s¯ t ( p), we can estimate as follows 1 2    j  2 2 2 2 f ≤2− f + f −− f − − B2σ (x¯k ( p))

Bs¯t ( p) (x¯k ( p))

(7.24)

≤

ϑ 2 h +C 2 k

i=0

B21−i σ (x¯k ( p))

j 

1

i=1

(2− j σ )m−2



B21−i σ (x¯k ( p))

B2−i σ (x¯k ( p))

|D N¯ k |2

  j ϑ 2  −j α ϑ 2 2 ϑ ≤ hk + Cc0 α hk (2 σ ) ≤ hk + C(α)c0 ϑ . 2 σ 2 i=1

185 Higher codimension integral currents

In the second line we have used the simple Morrey inequality       C 2 2 − f −− f  ≤ m−2 |D f |2  t B2t (x¯k ( p)) Bt (x¯k ( p)) B2t (x¯k ( p))  C ≤ m−2 |D N¯ k |2 . t B2t (x¯k ( p)) The constant C depends only upon the regularity of the underlying man¯ k , and, hence, can assumed independent of k. ifold M Since C(α) depends only on α, m and Q, for c0 chosen sufVciently small the latter inequality would contradict (7.19). Step (3). We collect the estimates (7.20) and (7.21) to infer the desired contradiction. We cover k with balls Bi := B20 pi ( pi ) such that B4 pi ( pi ) are disjoint, and deduce   (7.20) C(m) σ α  η m−2+α  pi ≤ |D N¯ k |2 ≤ C(m) 2 2 c 0 ϑhk i B pi (x¯k ( pi )) i α  α (7.10) C(m) σ σ ≤ |D N¯ k |2 ≤ C , 2 c0 ϑhk B 3 ϑ 2

where C(m) > 0 is a dimensional constant. We have used that the balls B pi (pM¯ k ( pi )) are pairwise disjoint by (7.21). Now note that ϑ and c0 are independent of σ , and therefore we can Vnally choose σ small enough to lead to a contradiction. 7.2.1 Persistence of Q-points Here we explain a simple instance of estimate (7.22), reporting the following theorem from [17]. Theorem 7.2 (Persistence of Q-points). For every δˆ > 0, there is s¯ ∈ ˆ > 0 with the fol]0, 12 [ such that, for every s < s¯ , there exists εˆ (s, δ) lowing property. If T is as in Theorem 3.9, E := E(T, C¯ 4 r (x)) < εˆ and !(T, ( p, q)) = Q at some ( p, q) ∈ C¯ r/2 (x), then the approximation f of Theorem 3.9 satisFes  ˆ m r 2+m E . G( f, Q [[η ◦ f ]])2 ≤ δs (7.25) Bsr ( p)

This theorem states that, in the presence of multiplicity Q points of the current, the Lipschitz (and therefore also the normal) approximations must have a relatively small L 2 norm, compared to the excess; or, as explained above, if in the normal approximation the excess is linked to the Dirichlet energy (for example this is the case of (EX)-cubes in the Whitney decomposition), the energy needs to be relatively large with respect to the L 2 norm, thus vaguely explaining the link to (7.22).

186 Emanuele Spadaro

Proof. By scaling and translating we assume x = 0 and r = 1; the choice of s¯ will be speciVed at the very end, but for the moment we impose s¯ < 14 . Assume by contradiction that, for arbitrarily small εˆ > 0, there are currents T and points ( p, q) ∈ C¯ 1/2 satisfying: E := E(T, C¯ 4 ) < εˆ , !(T, ( p, q)) = Q and, for f as in Theorem 3.9,  ˆ mE . G( f, Q [[η ◦ f ]])2 > δs (7.26) Bs ( p)

Set δ¯ = 14 and Vx η¯ > 0 (whose choice will be speciVed later). For a suitably small εˆ we can apply Theorem 3.10, obtaining a Dir-minimizing approximation w. If η¯ and εˆ are suitably small, we have  ˆ G(w, Q [[η ◦ w]])2 ≥ 34δ s m E , Bs ( p)

 and sup Dir(f), Dir(w)} ≤ CE. Then there exists p¯ ∈ Bs ( p) with G(w( p), ¯ Q [[η ◦ w( p)] ¯ ])2 ≥

3δˆ E, 4ωm

and, by the H¨older continuity in Theorem 3.2 (ii), we conclude g(x) := G(w(x), Q [[η ◦ w(x)]])

12 1 1 3δˆ ¯ s¯ κ ≥ δˆ E 2 , 2C E − 2 (C E) ≥ 4ω 2 m

(7.27)

where we assume that s¯ is chosen small enough in order to satisfy the last inequality. Setting h(x) := G( f (x), Q [[η ◦ f (x)]]), we recall that we have  |h − g|2 ≤ C ηE ¯ . Bs ( p)

  ˆ 1  Consider therefore the set A := h > 4δ E 2 . If η¯ is sufVciently small, we can assume that 1 |Bs ( p) \ A| < |Bs |. 8 Further, deVne A¯ := A∩K , where K is the set of Theorem 3.9. Assuming ¯ < 1 |Bs |. Let N be the εˆ is sufVciently small we ensure |Bs ( p) \ A| 4 ˆ

δE smallest integer such that N 64Qs ≥ 2s . Set

σi := s − i

δˆ E 64Qs

for i ∈ {0, 1 . . . , N },

187 Higher codimension integral currents

and consider, for i ≤ N − 1, the annuli Ci := Bσi ( p) \ Bσi+1 ( p). If εˆ is sufVciently small, we can assume that N ≥ 2 and σ N ≥ 4s . For at least one of these annuli we must have | A¯ ∩ Ci | ≥ 12 |Ci |. We then let σ := σi be the corresponding outer radius and we denote by C the corresponding annulus. Consider now a point x ∈ C ∩ A¯ and Tx be the slice T, p, x. Since let Q A¯ ⊂ K , for a.e. x ∈ A¯ we have Tx = i=1 [[(x, f i (x))]]. Moreover, there δˆ 1 2 exist i and j such that | f i (x)− f j (x)| ≥ Q G( f (x), [[η ◦ f (x)]])2 ≥ 4Q E (recall that x ∈ A¯ ⊂ A). When x ∈ C and the points (x, y) and (x, z) belong both to Bσ (( p, q)), we must have

2

δˆ E δˆ E δˆ E ≤ 8Q . ≤ σ8Qs |y − z|2 ≤ 4 σ 2 − σ − 64Qs Thus, for x ∈ A¯ ∩ C at least one of the points (x, f i (x)) is not contained in Bσ (( p, q)). We conclude therefore ¯ ≥ 1 |C| T (C¯ σ ( p) \ Bσ (( p, q))) ≥ |C ∩ A| 2

m

ωm m δˆ E σ − σ − 64Qs = 2

m

ωm m δˆ E . (7.28) ≥ σ 1 − 1 − 64Qsσ 2 . Since σ ≥ 4s , if Recall that, for τ sufVciently small, (1 − τ )m ≤ 1 − mτ 2 εˆ is chosen sufVciently small we can therefore conclude ωm σ m δˆ E ωm ˆ m−2 δ Eσ = c0 δˆ Eσ m−2 . ≥ 256Qsσ 1024Q (7.29) Next, by Theorem 3.9 and Theorem 3.10,  |Dw|2 m 1+γ1 ¯ + ηE ¯ + . (7.30) T (Cσ ( p)) ≤ Qωm σ + C E 2 Bσ ( p) T (C¯ σ ( p) \ Bσ ( p)) ≥

Moreover, as shown in [13, Proposition 3.10], we have  |Dw|2 ≤ CDir(w)σ m−2+2κ , Bσ ( p)

(7.31)

(for some constants κ and C depending only on m, n and Q; in fact the exponent κ is the one of Theorem 3.2 (ii)). Combining (7.29), (7.30) and (7.31), we conclude T (Bσ (( p, q))) ≤ Qωm σ m + η¯ E + C E 1+γ1 + C Eσ m−2+2κ − c0 σ m−2 δˆ E .

(7.32)

188 Emanuele Spadaro

Next, by the monotonicity formula, ρ → ρ −m T (Bρ (( p, q))) is a monotone function. Using !(T, ( p, q)) = Q, we conclude T (Bσ (( p, q))) ≥ Qωm σ m .

(7.33)

Combining (7.32) and (7.33) we conclude γ Cσ 2 + (η¯ + C E 1 )σ 2−m + Cσ 2κ ≥ c0 δˆ .

(7.34)

Recalling that σ ≤ s < s¯ , we can, Vnally, specify s¯ : it is chosen so that ˆ Combined with (7.27) this choice of s¯ C s¯ 2 + C s¯ 2κ is smaller than c20 δ. ˆ (7.34) becomes then depends, therefore, only upon δ. (η¯ + C E γ1 )σ 2−m ≥

c0 ˆ δ. 2

(7.35)

Next, recall that σ ≥ 4s . We then choose εˆ and η¯ so that (η+C ¯ εˆ γ1 )( 4s )2−m ≤ c0 ˆ δ. This choice is incompatible with (7.35), thereby reaching a contra4 diction: for this choice of the parameter εˆ (which in fact depends only upon δˆ and s) the conclusion of the theorem, i.e. (7.25), must then be valid.

8 Open questions We close this survey recalling some open problems concerning the regularity of area minimizing integer rectiVable currents. Some of them have been only slightly touched and would actually explain some of the complications that we met along the proof of the partial regularity result. For more open problems and comments, we suggest the reading of [1, 11]. One of the main, perhaps the most well-known, open problems is the uniqueness of the tangent cones to an area minimizing current, i.e. the uniqueness of the limit (ιx,r )" T as r → 0 for every x ∈ spt (T ). The uniqueness is known for two dimensional currents (cp. [35]), and there are only partial results in the general case (see [4, 30]). We have run into this issue in dealing with the step (C) of Section 2.7, because it is one of the possible reasons why a center manifold may be sufVcient in our proof. (A)

(B) A related question is that of the uniqueness of the inhomogeneous blowup for Dir-minimizing Q-valued functions. Also in this case the uniqueness is known for two dimensional domains (cp. [13], following ideas of [7]). Even if it does not play a role in the contradiction argument for the partial regularity, a positive answer to this question could indeed contribute to the solution of next two other major open problems.

189 Higher codimension integral currents

(C) It is unknown whether the singular set of an area minimizing current has always locally Vnite Hm−2 measure. This is the case for two dimensional currents (as proven by Chang [7]); note that in this result the uniqueness of the blowup Dir-minimizing map plays a fundamental role. (D) It is unknown whether the singular set of an area minimizing current has some geometric structure, e.g.. if it is rectiVable (i.e., roughly speaking, if it is contained in lower dimensional (m − 2)-dimensional submanifolds). Once again it is known the positive answer for two dimensional currents, where the singularities are known to be locally isolated, and the uniqueness of the tangent map is one of the fundamental steps in the proof.

We mention also the problem of Vnding more example of area minimizing currents, other than those coming from complex varieties or similar calibrations. Indeed, our understanding of the possible pathological behaviors of such currents is pretty much limited by the few examples we have at disposal. In particular, it would be extremely interesting to understand if there could be minimizing currents with weird singular set (e.g.., of Cantor type).

(E)

(F) Finally, we mention the problem of boundary regularity for higher codimension area minimizing currents, which to our knowledge is mostly open.

ACKNOWLEDGEMENTS . I am very grateful to A. Marchese, for reading a Vrst draft of these lecture notes and suggesting many precious improvements.

References [1] Some open problems in geometric measure theory and its applications suggested by participants of the 1984 AMS summer institute, In: “Geometric Measure Theory and the Calculus of Variations” (Arcata, Calif., 1984), J. E. Brothers (ed.), Proc. Sympos. Pure Math., Vol.44, Amer. Math. Soc., Providence, RI, 1986, 441–464. [2] W. K. A LLARD, On the Frst variation of a varifold, Ann. of Math. (2) 95 (1972), 417–491. [3] W. K. A LLARD, On the Frst variation of a varifold: boundary behavior, Ann. of Math. (2) 101 (1975), 418–446.

190 Emanuele Spadaro

[4] W. K. A LLARD and F. J. A LMGREN , J R ., On the radial behavior of minimal surfaces and the uniqueness of their tangent cones, Ann. of Math. (2) 113 (1981), 215–265. [5] F. J. A LMGREN , J R ., “Almgren’s big Regularity Paper”, World ScientiVc Monograph Series in Mathematics, Vol. 1, World ScientiVc Publishing Co. Inc., River Edge, NJ, 2000. [6] L. A MBROSIO, Metric space valued functions of bounded variation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 17 (1990), 439–478. [7] S. X U -D ONG C HANG, Two-dimensional area minimizing integral currents are classical minimal surfaces, J. Amer. Math. Soc. (4) 1 (1988), 699–778. [8] E. D E G IORGI, Su una teoria generale della misura (r − 1)dimensionale in uno spazio ad r dimensioni, Ann. Mat. Pura Appl. (4) 36 (1954), 191–213. [9] E. D E G IORGI, Nuovi teoremi relativi alle misure (r − 1)-dimensionali in uno spazio ad r dimensioni, Ricerche Mat. 4 (1955), 95–113. [10] E. D E G IORGI, Frontiere orientate di misura minima, Seminario di Matematica della Scuola Normale Superiore di Pisa, 1960-61, Editrice Tecnico ScientiVca, Pisa 1961. [11] C. D E L ELLIS, Almgren’s Q-valued functions revisited, In: “Proceedings of the International Congress of Mathematicians”, Volume III, Hindustan Book Agency, New Delhi, 2010, 1910–1933. [12] C. D E L ELLIS and E. S PADARO, Center manifold: a case study, Discrete Contin. Dyn. Syst. 31 (2011), 1249–1272. [13] C. D E L ELLIS and E. S PADARO, Q-valued functions revisited, Mem. Amer. Math. Soc. 211 (2011), vi+79. [14] C. D E L ELLIS and E. S PADARO, Regularity of area-minimizing currents II: center manifold, preprint, 2013. [15] C. D E L ELLIS and E. S PADARO, Regularity of area-minimizing currents III: blow-up, preprint, 2013. [16] C. D E L ELLIS and E. S PADARO, Multiple valued functions and integral currents, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) (2014), to appear. [17] C. D E L ELLIS and E. S PADARO, Regularity of area-minimizing currents I: gradient L p estimates, GAFA (2014), to appear. [18] H. F EDERER, “Geometric Measure Theory”, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969, xiv+676. [19] H. F EDERER and W. H. F LEMING, Normal and integral currents, Ann. of Math. (2) 72 (1960), 458–520.

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[20] W. H. F LEMING, On the oriented Plateau problem, Rend. Circ. Mat. Palermo (2) 11 (1962), 69–90. [21] M. F OCARDI , A. M ARCHESE and E. S PADARO, Improved estimate of the singular set of Dir-minimizing Q-valued functions, preprint 2014. [22] M. G ROMOV and R. S CHOEN, Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one, ´ Inst. Hautes Etudes Sci. Publ. Math. 76 (1992), 165–246. [23] R. H ARDT and L. S IMON, Boundary regularity and embedded solutions for the oriented Plateau problem, Ann. of Math. (2) 110 (1979), 439–486. [24] J. H IRSCH, Boundary regularity of Dirichlet minimizing Q-valued fucntions, preprint 2014. [25] J. J OST, Generalized Dirichlet forms and harmonic maps, Calc. Var. Partial Differential Equations 5 (1997), 1–19. [26] N. J. KOREVAAR and R. M. S CHOEN, Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom. 1 (1993), 561–659. [27] P. L OGARITSCH and E. S PADARO, A representation formula for the p-energy of metric space-valued Sobolev maps, Commun. Contemp. Math. 14 (2012), 10 pp. [28] E. R. R EIFENBERG, On the analyticity of minimal surfaces, Ann. of Math. (2) 80 (1964), 15–21. [29] T. R IVI E` RE, A lower-epiperimetric inequality for area-minimizing surfaces, Comm. Pure Appl. Math. 57 (2004), 1673–1685. [30] L. S IMON, Asymptotics for a class of nonlinear evolution equations, with applications to geometric problems, Ann. of Math. (2) 118 (1983), 525–571. [31] L. S IMON, “Lectures on Geometric Measure Theory”, Proceedings of the Centre for Mathematical Analysis, Australian National University, Vol. 3, Australian National University Centre for Mathematical Analysis, Canberra, 1983, vii+272. [32] L. S IMON, RectiFability of the singular sets of multiplicity 1 minimal surfaces and energy minimizing maps, In: “Surveys in Differential Geometry”, Vol. II (Cambridge, MA, 1993), Int. Press, Cambridge, MA, 1995, 246–305. [33] J. S IMONS, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. [34] E. S PADARO, Complex varieties and higher integrability of Dirminimizing Q-valued functions, Manuscripta Math. 132 (2010), 415–429.

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[35] B. W HITE, Tangent cones to two-dimensional area-minimizing integral currents are unique, Duke Math. J. 50 (1983), 143–160.

The regularity problem for sub-Riemannian geodesics Davide Vittone

Abstract. We study the regularity problem for sub-Riemannian geodesics, i.e., for those curves that minimize length among all curves joining two Vxed endpoints and whose derivatives are tangent to a given, smooth distribution of planes with constant rank. We review necessary conditions for optimality and we introduce extremals and the Goh condition. The regularity problem is nontrivial due to the presence of the so-called abnormal extremals, i.e., of certain curves that satisfy the necessary conditions and that may develop singularities. We focus, in particular, on the case of Carnot groups and we present a characterization of abnormal extremals, that was recently obtained in collaboration with E. Le Donne, G. P. Leonardi and R. Monti, in terms of horizontal curves contained in certain algebraic varieties. Applications to the problem of geodesics’ regularity are provided.

1 Introduction A sub-Riemannian manifold is a smooth, connected n-dimensional manifold M endowed with a smooth, bracket-generating sub-bundle  ⊂ T M (called horizontal), having constant rank r, and with a smooth metric g on . In these notes, we give a brief overview on the problem of the regularity of length minimizers, i.e., of the shortest (with respect to g) curves among all curves that join two Vxed endpoints and are horizontal, i.e., tangent to . We also present some results on the problem recently obtained, in the framework of Carnot groups, in collaboration with E. Le Donne, G. P. Leonardi and R. Monti [17, 18]. These notes are based on a course given by the author on the occasion of the ERC School Geometric Measure Theory and Real Analysis held at the Centro De Giorgi, Pisa, in October 2013. It is well-known (see e.g. the basic references [3, 4, 27]) that length minimizers are extremals, i.e., satisfy certain necessary conditions given

The author is supported by PRIN 2010-11 Project “Calculus of Variations” of MIUR (Italy), GNAMPA of INdAM (Italy), University of Padova, and Fondazione CaRiPaRo Project “Nonlinear Partial Differential Equations: models, analysis, and control-theoretic problems”.

194 Davide Vittone

by the Pontryagin Maximum Principle of Optimal Control Theory. Extremals may be either normal or abnormal: while normal extremals are always smooth, abnormal ones may develop singularities. Hence, the regularity problem for length minimizers is reduced to the regularity of abnormal minimizers. Let us spend a few words about the literature and the state of the art on the problem. We do not claim to be exhaustive and we refer to the beautiful introductions in [23, 30] for a more comprehensive account. It was originally claimed in [35] that length minimizing curves are smooth, all of them being normal extremals. The wrong argument relied upon an incorrect application of Pontryagin Maximum Principle, ignoring the possibility of abnormal extremals; see also [13]. A correction to [35] appeared in [36], where it was proved that minimizers in strong bracket-generating distributions are always normal and, hence, smooth. The Vrst example of a strictly abnormal length minimizer was provided by R. Montgomery in [26]. Other examples in the same vein are studied in [22,37]. Distributions of rank 2 are rich of abnormal geodesics: in [23], W. Liu and H. J. Sussmann introduced a class of abnormal extremals, called regular abnormal extremals, that are always locally length minimizing. Strictly abnormal length minimizers appear also in the setting of Carnot groups, see [11]. Notice, however, that all known examples of abnormal minimizers are smooth, so that the regularity problem is widely open. As we said, abnormal extremals may have singularities. In the paper [21], G. P. Leonardi and R. Monti developed an elaborate cuttingthe-corner technique (see also [2]) to show that, when the horizontal bundle satisVes a certain technical condition, length minimizers do not have corner-type singularities. In several interesting structures (among them, Carnot groups of rank 2 and nilpotency step at most 4), this is enough to conclude that length minimizers are smooth. More recently, R. Monti [29] was able to exclude certain singularities of higher order for length minimizers in structures satisfying the same condition introduced in [21]. Finally, a complete characterization of extremals in Carnot groups was recently obtained in [17, 18]. In particular, abnormal extremals in this setting are characterized as horizontal curves contained in certain algebraic varieties; the key tool here is represented by extremal polynomials. This allows for several applications; let us only mention the results discussed in these notes. First, one can give a very short proof of the regularity of length minimizers in Carnot groups of step not greater than 3 (a result Vrst proved in [38]). Second, we describe a new technique for proving the negligibility of the endpoints of abnormal extremals; for

195 The regularity problem for sub-Riemannian geodesics

the motivations behind this problem, which are only sketched in Remark 3.23, see [27, Section 10.2] and [2]. This technique cannot be applied to general Carnot groups; however, it is likely to work in many speciVc examples. A few words about the organization of these notes. In Section 2, we introduce the sub-Riemannian (or Carnot-Carath´eodory) distance. In Section 3, we derive the necessary conditions of extremality for length minimizers; the properties of normal and abnormal extremals are brieWy described in Sections 3.3 and 3.4. In Section 4, we introduce Carnot groups and present the characterization of extremals obtained in [17, 18]. In Section 5, we apply our results to prove the smoothness of minimizers in Carnot groups of step at most 3. Finally, in Section 6 we describe the technique connected with the negligibility problem for the endpoints of abnormal extremals.

2 The Carnot-Carath´eodory distance 2.1 DeHnition of Carnot-Carath´eodory distance A sub-Riemannian manifold is a smooth, connected n-dimensional manifold M endowed with a smooth, bracket-generating sub-bundle  ⊂ T M (called horizontal sub-bundle) of constant rank r and with a smooth metric g on . Without loss of generality, the regularity problem for length minimizers can be localized. Namely, we can assume that M = Rn and that the horizontal bundle  is generated by smooth, linearly independent vector Velds X 1 , . . . , X r which form an orthonormal system with respect to g. A Lipschitz continuous curve γ : [0, 1] → Rn is said to be horizontal if γ˙ (t) ∈ γ (t) for a.e. t ∈ [0, 1], i.e., if γ˙ (t) =

r 

h j (t)X j (γ (t)) for a.e. t ∈ [0, 1]

(2.1)

j=1

for suitable functions h = (h 1 , . . . , h r ) ∈ L ∞ ([0, 1], Rr ). We will refer to the functions h j as to the controls associated with γ . The length of γ is  1 |h(t)| dt . L(γ ) := 0

The fact that the length L is deVned by integrating |h(t)| := (h 1 (t)2 + · · · + h r (t)2 )1/2 corresponds to the fact that X 1 , . . . , X r are orthonormal.

196 Davide Vittone

DeHnition 2.1. The Carnot-Carath´eodory (CC) distance between x, y ∈ Rn is deVned as d(x, y) := inf {L(γ ) : γ is horizontal, γ (0) = x and γ (1) = y} . (2.2) The structure induced by the Carnot-Carath´eodory distance is often called sub-Riemannian because, intuitively speaking, the “allowed” directions form only a subspace of the whole tangent bundle.  1 1/2 Exercise 2.2. Given γ and h as above, deVne L 2 (γ ) := 0 |h(t)|2 dt ; prove that, for any x, y ∈ Rn , the CC distance d(x, y) is equal to d2 (x, y) := inf {L 2 (γ ) : γ is horizontal, γ (0) = x and γ (1) = y} . 2.2 The Chow-Rashevski theorem The family of curves in the right hand side of (2.2) might be empty (i.e., no horizontal curve joins x and y), hence d is not necessarily a distance. Consider, for instance, R3 with horizontal distribution generated by the vector Velds X 1 := (1, 0, 0) and X 2 := (0, 1, 0): clearly, in this case there is no horizontal curve joining the origin and the point (0, 0, 1). On the contrary, it is immediate to check that d is a distance whenever we can guarantee that any couple of points can be connected by horizontal curves. SufVcient conditions for connectivity are well-known; they are usually based on the following observation, which is a consequence of the Baker-Campbell-Hausdorff formula (see e.g. [40]). Here and in the sequel, we adopt the standard identiVcation between vector Velds and Vrst-order derivations. Fact. Given a point p ∈ Rn , two vector Velds X, Y and a positive real number t  1, one has e−tY e−t X etY et X ( p) = et

2 [X,Y ]

( p) + o(t 2 ) ,

(2.3)

where we deVne et X ( p) := c(t) as the curve c solving the problem c˙ = X (c), c(0) = p, and where the commutator (or bracket) [X, Y ] is the vector Veld X Y − Y X. Roughly speaking, if we are allowed to move along both X and Y , then we are also allowed to move in the direction of their commutator. This holds also for iterated brackets and suggests the following result, which we state without proof. Here and in the sequel, we denote by L(X 1 , . . . , X r ) the Lie algebra of vector Velds (with Lie product [·, ·]) generated by X 1 , . . . , X r .

197 The regularity problem for sub-Riemannian geodesics

Theorem 2.3. Assume that the bracket-generating condition rank L(X 1 , . . . , X r )(x) = n

∀x ∈ Rn

(2.4)

holds. Then, for any x, y ∈ Rn there exists a horizontal curve joining x and y; in particular, the Carnot-Carath´eodory distance d is an actual distance. Theorem 2.3 was proved independently by W. L. Chow [10] and P. K. Rashevski [33]; see also [8]. Condition (2.4) is also known as H¨ormander condition, as it was used by L. H¨ormander in the seminal paper [14] on hypoelliptic equations. In what follows, we will always assume that (2.4) is satisVed. 2.3 The Ball-Box Theorem In this section, we state the classical Ball-Box Theorem by A. Nagel, E. M. Stein and S. Wainger [32], that allows to compare (small) CC balls B(x, r) with suitable anisotropic boxes. See also [31]. If  ⊂ Rn is an open bounded set, then there exists an integer κ such that condition (2.4) is veriVed at every x ∈  by commutators of X 1 , . . . , X r with length at most κ (the length of a commutator [· · · [X j1 , X j2 ], X j3 ], . . . , X jm ] is by deVnition m). Let Y1 , . . . , Yq be a Vxed enumeration of all the commutators of length at most κ and let d(Yk ) ∈ {1, . . . , κ} denote the length of Yk . Given x ∈ Rn and a multi-index I = (i 1 , . . . , i n ) ∈ {1, . . . , q}n , deVne d(I) := d(Yi1 ) + · · · + d(Yin )  λI (x) := det col Yi1 (x) | Yi2 (x) | · · · | Yin (x) and the map E I (x, h) := eh 1 Yi1 +h 2 Yi2 +···+h n Yin (x),

h ∈ Rn .

Let us deVne the box BI (x, r) := {E I (x, h) : h ∈ Rn and max |h k |1/d(Yik ) < r} . k=1,...,n

We can then state the following result. Theorem 2.4. Let K ⊂  be a compact set; then, there exist positive numbers r, ˆ α, β, with β < α < 1, such that the following holds. If x ∈ K , r ∈ (0, r) ˆ and I are such that |λI (x)|r d(I ) >

1 2

max |λJ (x)|r d(J ) , J

(2.5)

198 Davide Vittone

then B(x, βr) ⊂ BI (x, αr) ⊂ B(x, r). In particular, there exists C = C(K ) > 0 such that d(x, y)  C|x − y|1/κ for any x, y ∈ K . Remark 2.5. As an important consequence, one can deduce from Theorem 2.4 that the topology induced by d is the standard one on Rn .

3 Length minimizers and extremals This section is devoted to the derivation of necessary conditions for length minimizing curves. Usually, such conditions are obtained by making use of the Pontryagin Maximum Principle of Optimal Control Theory; however, we will not directly refer to it. Our presentation is not meant to be exhaustive; the basic references [3, 4, 27] can be consulted for a more detailed account on these and related topics. 3.1 Length minimizers, existence and non-uniqueness DeHnition 3.1. A horizontal curve γ : [0, 1] → Rn is a length minimizer if it realizes the distance between its endpoints, i.e., if L(γ ) = d(γ (0), γ (1)). As a preliminary result, we prove the local existence of minimizers. Theorem 3.2. For any x ∈ Rn , there exists ρ > 0 with the following property: if d(x, y) < ρ, then there exists a length minimizer connecting x and y. Proof. Let x ∈ Rn be Vxed and let ρ > 0 be such that the CC ball B(x, ρ) is a bounded open subset of Rn . The existence of such a ρ is guaranteed by Remark 2.5. We are going to prove that, for any point y ∈ B(x, ρ), there exists a length minimizer from x to y. Let then x, ρ, y be as above and consider a sequence of horizontal curves γ k : [0, 1] → Rn , k ∈ N, such that γ k (0) = x,

γ k (1) = y

and

L(γ k ) → d(x, y) as k → ∞ .

In particular, for large k we have Im γ k ⊂ B(x, ρ)  Rn . Let h k : [0, 1] → Rr be the controls associated with γ k ; we can assume that, for any k, |h k | ≡ L(γ k ) is constant on [0, 1]. Thus, for large k, the Euclidean norm γ˙ k  L ∞ is bounded uniformly in k; by Ascoli-Arzel`a’s Theorem we deduce that, up to a subsequence, there exists a Lipschitz curve γ : [0, 1] → B(x, ρ) such that γ k → γ uniformly on [0, 1]. Now,

199 The regularity problem for sub-Riemannian geodesics

by the Dunford-Pettis theorem, up to a further subsequence we have that h k  h in L 1 ([0, 1], Rr ). For any t ∈ [0, 1] there holds  t r γ k (t) = h kj (s) X j (γ k (s)) ds . 0

j=1

Taking into account the uniform convergence of γ k and the weak convergence of h k , on passing to the limit as k → ∞ we get  t r γ (t) = h j (s) X j (γ (s)) ds , 0

j=1

i.e., the curve γ is horizontal with associated controls h. In particular we have γ (0) = x,

γ (1) = y

and

L(γ ) = h L 1  | lim inf h k  L 1 = d(x, y), k→∞

i.e., γ is a length minimizer connecting x and y. This concludes the proof. Unlike Riemannian geodesics, sub-Riemannian length minimizers are not unique, even locally. To illustrate this situation, we consider the subRiemannian Heisenberg group, i.e., the space R3 with horizontal distribution generated by the linearly independent vector Velds X 1 := ∂1 −

x2 ∂3 , 2

X 2 := ∂2 +

x1 ∂3 . 2

Notice that the bracket-generating condition is trivially satisVed because [X 1 , X 2 ] = ∂3 . Our aim it to describe length minimizers starting from the origin; a more detailed study can be found in [5]. It can be easily checked that a Lipschitz curve γ = (γ1 , γ2 , γ3 ) : [0, 1] → R3 is horizontal if and only if γ˙3 = −

γ2 γ1 γ˙1 + γ˙2 2 2

a.e. on [0, 1].

In particular, if c denotes the planar curve c(t) := (γ1 (t), γ2 (t)) and  is the planar region bounded by c and by the (oriented) segment σ joining c(1) to the origin, one can recover γ3 (1) as    x1

x1

x2 x2 γ3 (1) = − dx1 + dx2 = − dx1 + dx2 = dx1 ∧dx2 , 2 2 2 2 c c∪σ  where we have used Stokes’ theorem. Hence, the problem of connecting the origin (0, 0, 0) to (x, y, t) with a length minimizing horizontal curve

200 Davide Vittone

amounts to the problem of connecting (0, 0) to (x1 , x2 ) with the shortest planar curve enclosing (algebraic) area x3 . This is (a version of) Dido’s problem and it is well-known that, if x3 = 0, its solutions are arcs of circles. The corresponding horizontal curves are spirals which can be parametrized by arclength by the formulae ⎧ A(1 − cos ϕt) + B sin ϕt ⎪ ⎪ x1 (t) = ⎪ ⎪ ϕ ⎪ ⎪ ⎪ ⎨ −B(1 − cos ϕt) + A sin ϕt x2 (t) = (3.1) ⎪ ϕ ⎪ ⎪ ⎪ ⎪ ϕt − sin ϕt ⎪ ⎪ ⎩x3 (t) = − 2ϕ 2 for suitable (A, B) ∈ S 1 ⊂ R2 and ϕ = 0. If x3 = 0 we have instead the straight lines γ (t) = (Bt, At, 0). It can be proved that the spirals in (3.1) are length minimizing up to time t = 2π/ϕ, when they reach the point (0, 0, π/ϕ 2 ). In particular, for any ε > 0 there exists a family of length minimizers joining the origin 1 and (0, √ 0, ε): this family is parametrized by (A, B) ∈ S with the choice ϕ = π/ε. 3.2 First-order necessary conditions We want to derive necessary conditions for a horizontal curve to be length minimizing. To this end, we Vx a length minimizer γ : [0, 1] → Rn with associated optimal controls h. Without loss of generality, we may assume that γ (0) = 0 and that γ is parametrized by constant speed, i.e., that |h| = c a.e. on [0, 1]. In particular, by Exercise 2.2, γ is also a minimizer for the problem inf {L 2 (γ˜ ) : γ˜ is horizontal, γ˜ (0) = γ (0) and γ˜ (1) = γ (1)} . For any Vxed x ∈ Rn , let γx : [0, 1] → Rn be the solution of  γ˙x = h · X (γx ) γx (0) = x ,  where we write h · X (γx ) to denote the function rj=1 h j X j (γx ) deVned on [0, 1]. For any Vxed t ∈ [0, 1], let us deVne Ft : Rn → Rn by Ft (x) := γx (t) .

(3.2)

Notice that Ft is well deVned in a neighbourhood of the origin and that it is a diffeomorphism from such neighbourhood to its image.

201 The regularity problem for sub-Riemannian geodesics

Given another control k ∈ L ∞ ([0, 1], Rr ) we denote by qk the horizontal curve solving  q˙k = k · X (qk ) (3.3) qk (0) = 0 , Finally, given v ∈ L ∞ ([0, 1], Rr ), we deVne the (extended) endpoint map ϕv : R → Rn+1

1 ϕv (s) := F1−1 (qh+sv (1)), 0 (h + sv)2 . (3.4) The Vrst component of ϕv (s) is (up to a diffeomorphism) the endpoint of the horizontal curve qh+sv , while the last component is (the square of) its 2-length L 2 (qh+sv ). Lemma 3.3. If γ is length minimizing and parametrized by constant speed, then there exists ξ ∈ Rn+1 \ {0} such that ξ , ϕv (0) = 0 ∀v ∈ L ∞ ([0, 1], Rr ) .

(3.5)

Proof. Assume not: then, there exist v1 , . . . , vn+1 ∈ L ∞ ([0, 1], Rr ) such that the vectors ϕv 1 (0), . . . , ϕv n+1 (0) ∈ Rn+1 are linearly independent. Writing s · v := s1 v1 + · · · + sn+1 vn+1 , it follows that the map  : Rn+1 → Rn+1

  (s1 , . . . , sn+1 ) := F1−1 (qh+s·v (1)), L 2 (qh+s·v )2

is such that ∇(0) is invertible because ∂ (0) = ϕv i (0). In particular,  ∂si is an open map (in a neighbourhood of 0), hence one can Vnd s¯ ∈ Rn+1 such that the control h¯ := h + s¯1 v1 + · · · + s¯n+1 vn+1 satisVes F1−1 (qh¯ (1)) = F1−1 (qh (1)) L 2 (qh¯ )2 < L 2 (qh )2 . Since F1 is a diffeomorphism, if s¯ is close enough to 0, the Vrst equality above implies that qh¯ (1) = qh (1). This contradicts the minimality of γ = qh . Remark 3.4. An important role in the derivation of the necessary conditions in Theorem 3.6 will be played by the previous lemma. A key point in its proof is the fact that the extended endpoint map cannot be an open map in any neighbourhood of length minimizers. This suggests a sort of recipe to produce necessary conditions for optimality: in principle, any open mapping theorem might be used to derive necessary conditions. Also the Goh condition in the subsequent Theorem 3.20 is obtained by exploiting a suitable open mapping theorem.

202 Davide Vittone

Lemma 3.5. If v ∈ L ∞ ([0, 1], Rr ) is Fxed and ϕv is as in (3.4), then  1   1 ϕv (0) = J Ft (0)−1 (v · X (γ (t))) dt, 2 h(t), v(t) dt (3.6) 0 0 n ∈ R × R, where J Ft is the Jacobian matrix of Ft and, again, v · X = v1 X 1 + · · · + vr X r . Proof. Let s ∈ R be Vxed and, for any t ∈ [0, 1], deVne x h+sv (t) := Ft−1 (qh+sv (t)); equivalently, qh+sv (t) = Ft (x h+sv (t)) .

(3.7)

In particular, the Vrst n components in the deVnition of ϕv (s) are equal to x h+sv (1). We can differentiate (3.7) with respect to t to obtain (h + sv) · X (qh+sv ) = h · X (qh+sv ) + J Ft (x h+sv )x˙h+sv , hence

x˙h+sv = s J Ft (x h+sv )−1 [v · X (qh+sv )] .

It follows that  xh+sv (t) = s 0

t

J Fτ (x h+sv (τ ))−1 [v · X (Fτ (x h+sv (τ )))] dτ ,

i.e., ϕv (0)

   1 ∂ x h+sv (1)  = h(τ ), v(τ ) dτ  ,2 ∂s 0 s=0   1  1 −1 J Fτ (x h (τ )) [v · X (Fτ (x h (τ )))] dτ, 2 h(τ ), v(τ )dτ . = 

0

0

The desired equality (3.6) easily follows on noticing that xh (τ ) = Fτ−1 (qh (τ )) = Fτ−1 (γ (τ )) = 0. We can now pass to the main result of this section. Theorem 3.6 (First-order necessary conditions). Let γ : [0, 1] → Rn be a length minimizer with γ (0) = 0 and with associated controls h; assume that γ is parametrized by constant speed, i.e., |h| ≡ c. Then, there exist ξ0 ∈ {0, 1} and ξ ∈ Lip([0, 1], Rn ) such that

203 The regularity problem for sub-Riemannian geodesics

(i) (ξ(t), ξ0 )  = 0 for any t ∈ [0, 1]; (ii) for any j = 1, . . . , r, the equality ξ0 h j + ξ, X j (γ ) = 0 holds a.e. on [0, 1];  (iii) ξ˙ = −( rj=1 h j J X j (γ ))T ξ a.e. on [0, 1], where J X j denotes the n × n Jacobian matrix of X j : Rn → Rn and the superscript T denotes matrix transposition. Proof. Let ξ ∈ Rn+1 \ {0} be as in Lemma 3.3; write ξ =: (ξ(0), ξ0 /2) ∈ Rn ×R. Using Lemma 3.5, we deduce from (3.5) the following necessary condition:  0=

1

r 

1

j=1 r 

0

 =

0

  v j (t) ξ(0), J Ft (0)−1 (X j (γ (t))) + ξ0 h j dt   v j (t) [J Ft (0)−1 ]T ξ(0), X j (γ (t)) + ξ0 h j dt

j=1 ∞

∀v ∈ L ([0, 1], Rr ) . Upon deVning ξ(t) := [J Ft (0)−1 ]T ξ(0), the Fundamental lemma of the Calculus of Variations immediately implies statement (ii). Statement (i) is clearly true if ξ0 = 0 (notice that, in this case, one can also normalize ξ to have ξ0 = 1); on the contrary, if ξ0 = 0 we have ξ(0)  = 0, hence ξ(t) = 0 for all t ∈ [0, 1] because J Ft (0)−1 is invertible. Hence, also (i) is proved. We are left with statement (iii). By deVnition of ξ(t), we have ξ(0) = J Ft (0)T ξ(t) and, on differentiating with respect to t,   d T (3.8) 0= J Ft (0) ξ(t) + J Ft (0)T ξ˙ (t) a.e. on [0, 1] . dt Let us compute 1 2  r    d d =J h j (t)X j (Ft (x))  J Ft (0) = J Ft (x) dt dt x=0 x=0 j=1 =

r 

h j (t) J X j (Ft (0)) J Ft (0)

j=1

=

 r j=1

 h j (t) J X j (γ (t)) J Ft (0) a.e. on [0, 1] .

204 Davide Vittone

Recalling (3.8) and the fact that J Ft (0)T is invertible, we obtain  T r ξ˙ (t) = − h j (t) J X j (γ (t)) ξ(t) for a.e. t ∈ [0, 1] , j=1

as desired. DeHnition 3.7. A horizontal curve γ : [0, 1] → Rn with γ (0) = 0 and with associated controls h is said to be an extremal if there exist ξ0 ∈ {0, 1} and ξ ∈ Lip([0, 1], Rn ) such that statements (i), (ii) and (iii) in Theorem 3.6 hold. The function ξ is called dual curve (or dual variable). If ξ0 = 1, we say that γ is a normal extremal. If ξ0 = 0, we say that γ is an abnormal extremal. Theorem 3.6 states that length minimizers parametrized by constant speed are also extremals; on the contrary, there exist extremals that are not minimizers, see Section 3.5. We do not require extremals to be parametrized by constant speed because this is automatically satisVed for normal extremals (see Exercise 3.11), while for abnormal extremals the parametrization plays essentially no role (see Exercise 3.17). We will review the main properties of normal and abnormal extremals in Sections 3.3 and 3.4; now, a few observations are in order. Remark 3.8. An extremal γ might be normal and abnormal at the same time, in the sense that it could possess two different dual curves that make γ normal and abnormal. An example of this phenomenon is given in Exercise 4.5. An extremal which is normal but not abnormal is called strictly normal; on the contrary, we call strictly abnormal an extremal which is abnormal but not also normal. Exercise 3.9. Prove that, if γ is strictly normal, then it possesses a unique dual curve ξ(t). Hint: assume that ξ1 (t), ξ2 (t) are dual curves making γ normal; prove that γ is abnormal with associated dual curve ξ1 − ξ2 . Theorem 3.6 possesses also an Hamiltonian formulation. DeVne the Hamiltonian H (x, ξ ) :=

r  X j (x), ξ 2 j=1

Then, the following result holds.

x, ξ ∈ Rn .

205 The regularity problem for sub-Riemannian geodesics

Exercise 3.10. If γ is a normal extremal with dual variable ξ , then the couple (γ , ξ ) solves the system of Hamiltonian equations ⎧ 1 ∂H ⎪ ⎪ (γ , ξ ) ⎨ γ˙ = − 2 ∂ξ ⎪ ⎪ ⎩ ξ˙ = 1 ∂ H (γ , ξ ) . 2 ∂x If γ is an abnormal extremal with dual variable ξ , then H (γ , ξ ) ≡ 0. 3.3 Normal extremals In this section we deal with basic properties and facts about normal extremals. We begin with the following exercise. Exercise 3.11. Let γ be a normal extremal; then, γ is parametrized by constant speed. Hint: use Exercise 3.10 and the fact that, if h denotes the controls associated with γ , then |h(t)|2 = H (γ (t), ξ(t)). The most important result in this subsection is the following one. Proposition 3.12. Normal extremals are C ∞ smooth. Proof. Let γ : [0, 1] → Rn be a normal extremal with associated controls h and dual curve ξ . Using (2.1) and (ii), (iii) in Theorem 3.6 we easily obtain the following chain of implications (ii)

γ , ξ ∈ C 0 ([0, 1]) (⇒ h j ∈ C 0 ([0, 1]) ∀ j = 1, . . . , r (2.1),(iii)

(ii)

(2.1),(iii)

(ii)

(⇒ γ , ξ ∈ C 1 ([0, 1]) (⇒ h j ∈ C 1 ([0, 1]) ∀ j = 1, . . . , r (⇒ γ , ξ ∈ C 2 ([0, 1]) (⇒ h j ∈ C 2 ([0, 1]) ∀ j = 1, . . . , r (⇒ . . .

Exercise 3.13. Prove that, if γ is a normal extremal with dual curve ξ , then condition (ii) in Theorem 3.6 holds on the whole interval [0, 1] (and not only almost everywhere). The following results, as well as the Proposition 3.12, show that normal minimizers/extremals share several common features with Riemannian geodesics. Remark 3.14. When r = n (i.e., the CC structure is indeed Riemannian), any length minimizer/extremal γ is strictly normal. Otherwise, there would exist a dual curve ξ such that ξ, X j (γ ) = 0 for any j = 1, . . . , n. Since X 1 , . . . , X n now form a basis of Rn , we obtain that ξ ≡ 0, which contradicts (i) in Theorem 3.6.

206 Davide Vittone

Exercise 3.15. Assume again that we are in the Riemannian case r = n. Then, by (2.1) and (ii) in Theorem 3.6, there is a natural way of identifying γ˙ , h and ξ , in the sense that any of the three uniquely determines the others. Prove that equation (iii) in Theorem 3.6 corresponds to the ODE of Riemannian geodesics. The following important result is a special case of more general results in Optimal Control Theory, see for instance [7,13,20] and [23, Appendix C]. Theorem 3.16. Every normal extremal is locally length minimizing. On the contrary, strictly abnormal extremals might not be length minimizers, see Section 3.5. 3.4 Abnormal extremals By Theorem 3.6 (ii), an abnormal extremal γ and its dual variable ξ satisfy (3.9) ξ, X j (γ ) = 0 on [0, 1] ∀ j = 1, . . . , r . The compact notation ξ ⊥ γ will often be used to abbreviate the previous formula. When dealing with abnormal extremals, it is not necessary to require that they are parametrized by constant speed; this is justiVed by the following fact. Exercise 3.17. Assume that γ˜ : [0, 1] → Rn is an abnormal extremal parametrized by constant speed and with dual curve ξ˜ . Let γ be a different parametrization of the same curve; namely, let γ := γ˜ ◦ f for an increasing, Lipschitz continuous homeomorphism f : [0, 1] → [0, 1]. Then, γ satisVes (i), (ii) and (iii) in Theorem 3.6 with ξ := ξ˜ ◦ f . Exercise 3.18. Prove that, if γ is an abnormal extremal with dual curve ξ , then condition (ii) in Theorem 3.6 holds on the whole interval [0, 1] (and not only almost everywhere). Abnormal extremals are often introduced in the literature as singular points of the endpoint map; a few comments on this viewpoint are in order. Going back to Section 3.2, let γ : [0, 1] → Rn be an extremal with γ (0) = 0 and associated optimal controls h ∈ L ∞ ([0, 1], Rr ). DeVne the endpoint map End : L ∞ ([0, 1], Rr ) → Rn by End(k) := qk (1),

k ∈ L ∞ ([0, 1], Rr ) ,

207 The regularity problem for sub-Riemannian geodesics

the curve qk being deVned as in (3.3). For any v ∈ L ∞ ([0, 1], Rr ), the map ϕv (s) in (3.4) can then be rewritten as   ϕv (s) = F1−1 ◦ End(h + sv), L 2 (qh+sv )2 , where the diffeomorphism F1 is deVned as in (3.2). Now, if γ is an abnormal extremal, then the vector ξ = (ξ(0), ξ0 /2) ∈ Rn × R provided by Lemma 3.3 is such that ξ0 = 0. Hence (again by Lemma 3.3), the vector ξ(0) = 0 is such that   d  −1 ξ(0) ⊥ ∀v ∈ L ∞ ([0, 1], Rr ) . F ◦ End(h + sv)  ds 1 s=0 Since F1−1 is a diffeomorphism, there exists also a vector η  = 0 such that   d  η⊥ = dEnd(h)[v] ∀v ∈ L ∞ ([0, 1], Rr ) , End(h + sv)  ds s=0 where dEnd(h)[v] denotes the differential of End at h in direction v. In particular, the image of dEnd(h) does not contain the vector η; equivalently, h is a point where the differential of the endpoint map is not surjective. We have proved that (the controls associated with) abnormal extremals are singular points of End; the following exercise shows that the converse is also true. Exercise 3.19. Prove that, if the differential of the endpoint map End is not surjective at some controls h associated with an horizontal curve γ , then γ is an abnormal extremal. As already pointed out in Remark 3.8, an extremal might be normal and abnormal at the same time. By Proposition 3.12 any minimizer/extremal is C ∞ smooth unless it is strictly abnormal; hence, the relevant curves in the regularity problem for length minimizers are precisely the strictly abnormal ones. For such minimizers, a further necessary condition, the so-called Goh condition, can be proved. Theorem 3.20 (Goh condition). Let γ : [0, 1] → Rn be a strictly abnormal length minimizer. Then, there exists an associated dual curve ξ that satisFes ξ, [X i , X j ](γ ) = 0 on [0, 1] for any i, j = 1, . . . , r.

(3.10)

We refer to [4, Chapter 20] for the proof of Theorem 3.20. The proof is in the spirit of Remark 3.4: if (3.10) does not hold for any dual curve ξ , then a suitable open mapping theorem allows to conclude that a certain mapping of endpoint-type is open at γ , contradicting its minimality.

208 Davide Vittone

Corollary 3.21. If the horizontal distribution X 1 , . . . , X r is of step 2, i.e., if dim span {X i , [X i , X j ] : i, j ∈ {1, . . . , r}}(x) = n

∀x ∈ Rn ,

then any length minimizer is C ∞ smooth. Proof. Assume by contradiction that there exists a length minimizer γ that is not of class C ∞ ; then, by Proposition 3.12, γ is strictly abnormal. By (3.9) and Theorem 3.20, there exists a dual variable ξ that is orthogonal (at points of γ ) to X i and [X i , X j ] for any i, j ∈ {1, . . . , r}. Since, by assumption, these elements generates all the tangent space at any point, then we have necessarily ξ ≡ 0, which contradicts Theorem 3.6 (i). We stress the fact that the minimality assumption is crucial in Theorem 3.20. In general, (3.10) might not hold for strictly abnormal extremals, with the following remarkable exception concerning general abnormal extremals in structures with rank 2. Remark 3.22. If the horizontal distribution has rank r = 2, then any abnormal extremal γ : [0, 1] → Rn and any associated dual curve ξ satisfy ξ(t), [X 1 , X 2 ](γ (t)) = 0 ∀t ∈ [0, 1] (3.11) Let us prove (3.11); we claim that it is enough to show that ξ(t), [X 1 , X 2 ](γ (t)) = 0 for a.e. t ∈ [0, 1] such that γ˙ (t) exists and γ˙ (t)  = 0 .

(3.12)

Indeed, (3.12) and the continuity of ξ imply (3.11) for any γ such that γ˙  = 0 a.e. on [0, 1]; for instance, whenever γ is parametrized by constant speed. For different parametrizations of γ , it is enough to reason as in Exercise 3.17. Let us prove (3.12). By equation (iii) in Theorem 3.6 and the abnormality of γ , we get d ξ, X 1 (γ ) dt = −(h 1 J X 1 (γ ) + h 2 J X 2 (γ ))T ξ, X 1 (γ ) + ξ, J X 1 (γ )[γ˙ ] = −ξ, (h 1 J X 1 (γ ) + h 2 J X 2 (γ ))[X 1 (γ )] + ξ, J X 1 (γ )[h 1 X 1 (γ ) + h 2 X 2 (γ )] = h 2 ξ, −J X 2 (γ )[X 1 (γ )] + J X 1 (γ )[X 2 (γ )] a.e. on [0, 1]. = −h 2 ξ, [X 1 , X 2 ](γ )

0=

209 The regularity problem for sub-Riemannian geodesics

With similar computations one gets d 0 = ξ, X 2 (γ ) = h 1 ξ, [X 1 , X 2 ](γ ) a.e. on [0, 1]. dt In particular, if t ∈ [0, 1] is such that γ˙ (t) = 0, then either h 1 (t)  = 0 or h 2 (t)  = 0, and this is enough to obtain (3.12). As done before, for notational convenience we write ξ ⊥ (∪[, ])γ whenever the Goh condition holds for the couple (γ , ξ ), to mean that the dual variable ξ is orthogonal to both horizontal vectors and brackets of horizontal vector Velds. With a slight change of notation, we could also introduce the time-dependent 1-form ξ ∗ (t) := ξ1 (t)dx1 + · · · + ξn (t)dxn

(3.13)

∗ ⊥ ⊥ and write ξ ∗ ∈ ⊥ γ (for abnormal extremals) or ξ ∈ γ ∩ [, ]γ ∗ (when the Goh condition is in force). The 1-form ξ is going to appear again later in these notes. Remark 3.23. Another important fact about abnormal minimizers has been proved in [1] (see also [34]) in connection with the smoothness problem for the CC distance d: if the horizontal vectors X 1 , . . . , X r are analytic, then the set  of point in Rn that can be connected to the origin (or to any other base point) with abnormal length minimizers is a closed set with empty interior. An important open question is the following Morse-Sard problem for the endpoint map: does  have measure zero? We refer to [27, Section 10.2] and to the recent preprint [2] for more detailed discussions on this and other topics.

3.5 An interesting family of extremals An interesting sub-Riemannian structure was proposed by A. Agrachev and J. P. Gauthier during the meeting “Geometric control and subRiemannian geometry” held in Cortona in May 2012. Consider the CC structure of rank 2 induced on R4 by the vector Velds X 1 (x) := ∂1 + 2x2 ∂3 + x32 ∂4 ,

X 2 (x) := ∂2 − 2x1 ∂3 .

It can be easily checked that the bracket-generating condition holds and that, for any α ∈ R, the curve γ α (t) := (t, α|t|, 0, 0), t ∈ R, is a strictly abnormal extremal with dual curve ξ(t) = (0, 0, 0, 1). It is fairly easy to show that γ 0 is a minimizer; R. Monti has proved with a cutting-the/ {0, 1, −1}. Using corner technique that γ α is not a minimizer when α ∈ a different and much simpler argument, the remaining case α = ±1 was recently settled in [19], where it is also proved that all length minimizers in the CC structure under consideration are smooth. Exercise 3.24. Prove that γ 0 is uniquely length minimizing.

210 Davide Vittone

4 Carnot groups 4.1 StratiHed groups In this section we are going to describe a few basic facts on stratiVed groups. Recall that the Lie algebra g associated with a Lie group G is deVned as the Lie algebra of left-invariant vector Velds on G. A vector Veld X on G is said to be left-invariant if X ( p) = d& p (X (0)) ∀ p ∈ G , where d& p denotes the differential of the left-translation & p (z) = p · z by p, · denotes the group product and 0 denotes the identity of G. Equivalently, X is left-invariant if (X f )(& p (x)) = X ( f ◦ & p )(x) for any p, x ∈ G and any f ∈ C ∞ (G). DeHnition 4.1. A stratiFed group G is a connected, simply connected and nilpotent Lie group whose Lie algebra g admits a stratiFcation, i.e., a decomposition g = V1 ⊕ V2 ⊕ · · · ⊕ Vs with the properties that Vi = [V1 , Vi−1 ] for any i = 2, . . . , s and [V1 , Vs ] = {0}. A few comments are in order: • the Lie algebra g is nilpotent of step s; • one can easily see that [Vi , V j ] ⊂ Vi+ j for any i, j  1 such that i + j  s; • if i + j  s + 1, then [Vi , V j ] = {0}. Moreover, the exponential map exp : g → G induces a diffeomorphism between G and Rn ≡ g, n being the dimension of g. However, in the sequel we will identify G with Rn by means of a different set of coordinates, the so-called exponential coordinates of the second-type (see (4.1) and (4.2) below). Let us Vx an adapted basis of g, i.e., a basis X 1 , . . . , X n whose order is coherent with the stratiVcation: X , . . . , X , X , . . . , X r2 , X r2 +1 , . . . . . . . . . . . . , X n . 5 1 67 8r 5 r+1 67 5 67 8 8 5 67 8 basis of V1

basis of V2

basis of V3

basis of Vs

The integer r2 := dim V1 + dim V2 will be used also in the sequel. We can then identify G with Rn by introducing exponential coordinates of the second type Rn # (x1 , . . . , xn ) ←→ exp(xn X n ) · exp(xn−1 X n−1 ) · · · exp(x1 X 1 ) ∈ G (4.1)

211 The regularity problem for sub-Riemannian geodesics

or, equivalently, by using Wows of vector Velds Rn # (x1 , . . . , xn ) ←→ e x1 X 1 ◦ · · · ◦ e xn−1 X n−1 ◦ e xn X n (0) ∈ G .

(4.2)

As a matter of fact (see e.g. [18]), one can prove that in these coordinates X 1 = ∂1  X i (x) = ∂i + nj=r+1 f i j (x)∂ j

∀i = 2, . . . , r

(4.3)

for suitable analytic functions f i j : Rn → R. The stratiVcation of g allows to deVne a family of intrinsic dilations on G. For any i = 1, . . . , r, let us deVne its degree d(i) ∈ {1, . . . , s} by d(i) = k ⇐⇒ X i ∈ Vk . One can deVne a one-parameter family of dilations on g in the following way. For any r > 0, let δr : g → g be the unique linear map such that δr (X i ) = r d(i) X i . Then, by the stratiVcation assumption, δr is a Lie algebra isomorphism. One can also deVne dilations on the group (in coordinates) by δr (x1 , . . . , xn ) := (r x1 , . . . , r d(i) xi , . . . , r s xn ) . Clearly, δr : G → G deVnes a one-parameter family of group isomorphisms. Example 4.2. The Heisenberg group (see also Section 3.1, where it is presented in a different set of coordinates) is the stratiVed group associated with the Lie algebra of step 2 g := V1 ⊕ V2 , where V1 = span {X 1 , X 2 }, V2 = span {X 3 } and [X 2 , X 1 ] = X 3 ,

[X 3 , X 1 ] = [X 3 , X 2 ] = 0 .

The Heisenberg group can be represented in exponential coordinates of the second type as R3 with X 1 = ∂1 ,

X 2 = ∂2 − x1 ∂3 ,

X 3 = ∂3 .

Group dilations read as δr (x1 , x2 , x3 ) = (r x1 , r x2 , r 2 x3 ). Example 4.3. The Engel group is the stratiVed group associated with the Lie algebra of step 3 g := V1 ⊕ V2 ⊕ V3 , where V1 = span {X 1 , X 2 }, V2 = span {X 3 }, V3 = span {X 4 } and [X 2 , X 1 ] = X 3 , [X 3 , X 1 ] = X 4 , [X 3 , X 2 ] = [X 4 , X 1 ] = [X 4 , X 2 ] = [X 4 , X 3 ] = 0 .

212 Davide Vittone

The Engel group can be represented in exponential coordinates of the second type as R4 with X 1 = ∂1 ,

X 2 = ∂2 − x1 ∂3 +

x12 ∂4 , 2

X 3 = ∂3 − x1 ∂4 ,

X 4 = ∂4 .

Group dilations read as δr (x1 , x2 , x3 , x4 ) = (r x1 , r x2 , r 2 x3 , r 3 x4 ). 4.2 Carnot groups StratiVed groups can be endowed with a canonical CC structure induced by a basis X 1 , . . . , X r of the Vrst layer V1 . Notice that the horizontal sub-bundle  := V1 is left-invariant and bracket-generating (by the stratiVcation assumption), hence the CC distance d is well deVned. We refer to [15] for a metric characterization of Carnot groups and to [16] for an introduction to sub-Riemannian geometry on groups. Exercise 4.4. Prove that, for any p, x, y ∈ G and any r > 0, there holds d( p · x, p · y) = d(x, y) and d(δr x, δr y) = rd(x, y) . Exercise 4.5. Prove that the horizontal curve γ (t) = (0, t, 0, 0) in the Engel group (represented in the coordinates of Example 4.3) is an extremal that is normal and abnormal at the same time. Our interest in Carnot groups is motivated by the well-known fact that the tangent metric space (in the Gromov-Hausdorff sense) to a CC space at a “generic” point is a Carnot group: roughly speaking, Carnot groups are the inVnitesimal models of CC spaces. See e.g. [6, 24, 25]. 4.3 The dual curve and extremal polynomials Let γ : [0, 1] → G be an extremal with associated controls h ∈ L ∞ ([0, 1], Rr ) and dual curve ξ ∈ Lip([0, 1], Rn ); assume that γ (0) = 0. Recall that ξ induces a time-dependent 1-form ξ ∗ as in (3.13); we are going to write ξ ∗ in a different system of coordinates for 1-forms. The group structure allows to deVne a frame θ1 , . . . , θn of left-invariant 1-forms, dual to the adapted basis X 1 , . . . , X n , by imposing that θi (X j ) = δi j

on G ,

(4.4)

δi j denoting the Kronecker delta. We can therefore deVne λ∈Lip([0,1],Rn ) by imposing that ξ ∗ (t) = ξ1 (t)dx1 + · · · + ξn (t)dxn   = λ1 (t)θ1 + · · · + λn (t)θn (γ (t)) ∀ t ∈ [0, 1] .

(4.5)

213 The regularity problem for sub-Riemannian geodesics

We use the term dual curve also for the function λ. One can immediately notice that, by (4.4), statement (iii) in Theorem 3.6 is equivalent to ξ0 h i + λi = 0 a.e. on [0, 1],

∀ i = 1, . . . , r .

(4.6)

Moreover, the differential equation (iii) of Theorem 3.6 is equivalent to the following ODE for λ (we refer to [17, Theorem 2.6] for details). For any i = 1, . . . , n, there holds λ˙ i = −

r  n 

cikj h j λk

a.e. on [0, 1] ,

(4.7)

j=1 k=1

where the constants cikj are the structure constants of the Lie algebra g deVned by [X i , X j ] =

n 

cikj X k

∀ i, j = 1, . . . , n .

k=1

Exercise 4.6. Prove the implication d(k)  = d(i) + d( j) ⇒ cikj = 0 ∀i, j, k = 1, . . . , n .

(4.8)

Hint: recall that [X i , X j ] ∈ Vd(i)+d( j) . Deduce, as a consequence, that (4.7) is equivalent to λ˙ i = −

r 



j=1

k=1,...,n d(k)=d(i)+1

cikj h j λk

a.e. on [0, 1] .

(4.9)

From the technical viewpoint, the main achievement of [17, 18] is an explicit formula for the dual curve λ as a function of γ , see Theorem 4.11 below. This is obtained through the integration of the ODE (4.7), which is in turn based on the following result. Lemma 4.7. Let γ : [0, 1] → G be an extremal with γ (0) = 0; let h ∈ L ∞ ([0, 1], Rr ) be the associated controls and λ ∈ Lip([0, 1], Rn ) be its dual curve. Suppose that there exist functions Pi ∈ C 1 (G), i = 1, . . . , n, such that Pi (0) = λi (0) and

X j Pi =

n 

ckji Pk on G

(4.10)

k=1

for any i, j = 1, . . . , n. Then, for any i = 1, . . . , n, there holds λi (t) = Pi (γ (t)) ∀ t ∈ [0, 1].

(4.11)

214 Davide Vittone

Proof. The proof is based on a reverse-order inductive argument on i; we start by proving (4.11) for i = n. Since X n ∈ Vs is in the kernel of g, we have [X j , X n ] = 0 for any j = 1, . . . , n, i.e., ckjn = −cnk j = 0. In particular, by (4.7) and (4.10)   • λ˙ n = − rj=1 nk=1 cnk j h j λk = 0, hence λn is constant on [0, 1];  • for any j = 1, . . . , n, X j Pn = nk=1 ckjn Pk = 0, hence Pn is constant on G. Since, by assumption, Pn (0) = λn (0), we obtain that λn (t) = Pn (γ (t)) for any t ∈ [0, 1]. now that λk = Pk (γ ) for any k  i + 1; recalling that γ˙ = Assume r h X (γ ), we have j j j=1 r r  n   d h j X j Pi (γ ) = h j ckji Pk (γ ) (Pi ◦ γ ) = dt j=1 j=1 k=1 (4.8)

=

r 



j=1

k=1,...,n d(k)=d(i)+1

h j ckji Pk (γ ) .

We can now use the inductive assumption together with the equality ckji = −cikj to get r  d (Pi ◦ γ ) = − dt j=1



(4.9) h j cikj λk (γ ) = λ˙ i .

k=1,...,n d(k)=d(i)+1

In particular, the Lipschitz functions λi and Pi ◦ γ have the same derivative and, by assumption, they coincide at time t = 0. This is sufVcient to conclude the validity of (4.11). The integration of the dual variable λ is thus reduced to the search for functions Pi satisfying (4.10); these functions are provided by the extremal polynomials introduced below in DeVnition 4.8. Let us introduce some preliminary notation. Given a multi-index α = (α1 , . . . , αn ) ∈ Nn and x ∈ Rn ≡ G, we write x α = x1α1 x2α2 · · · xnαn |α| = α1 + · · · + αn α! = α1 ! α2 ! · · · αn ! . For the sake of precision: we agree that 0 ∈ N, hence the null multi-index α = 0 is admissible. If x = 0 and α = 0, we agree that x α = 1.

215 The regularity problem for sub-Riemannian geodesics

DeHnition 4.8. For any v ∈ Rn and i = 1, . . . , n, we deVne the extremal polynomial Piv : G → R by Piv (x) =

n  (−1)|α| k ciα vk x α , α! n α∈N k=1

(4.12)

k denote the generalized structure constants of g where the symbols ciα deVned by

[· · · [X i , X 1 ], X 1 ], . . . X 1 ], X 2 ], . . . X 2 ], X 3 ], . . . ] . . . ] = 67 8 5 67 8 5 α1 times

α2 times

n 

k ciα Xk .

k=1

Exercise 4.9. Prove that the summation in (4.12) is Vnite and, more precisely, that Piv is a polynomial of both degree and homogeneous degree (see e.g. [17, Remark 4.2])  at most s − d(i). Hint: deVne d(α) := nj=1 αi d(i) and prove the implication k = 0. d(k) = d(i) + d(α) ⇒ ciα

As already mentioned, extremal polynomials satisfy (4.10) in Lemma 4.7. Theorem 4.10. For any v ∈ Rn and i = 1, . . . , n, the extremal polynomials satisfy Piv (0) = vi

and

X j Piv =

n 

ckji Pkv on G.

(4.13)

k=1

While the Vrst equality in (4.13) can be easily checked, the formulae for the derivatives of the Piv ’s are not trivial at all. Their proof is however beyond the scopes of these notes. In the framework of free Carnot groups, the second equality in (4.13) was Vrst proved in [17] as a consequence of certain algebraic identities obtained along the proof of [17, Theorem 4.6]. The latter is nothing but Theorem 4.11 below (in the special case of free groups), but its proof follows a completely different line from the one presented here, being based on explicit formulae for the horizontal vector Velds (see [12]) rather than on Lemma 4.7. For general Carnot groups, the proof of (4.13) was achieved in [18] with an argument of a differentialgeometric Wavour involving also non-trivial algebraic identities. Lemma 4.7 and Theorem 4.10 have the following, immediate consequence. Theorem 4.11. Let γ : [0, 1] → G be an extremal with γ (0) = 0; let λ ∈ Lip([0, 1], Rn ) be an associated dual curve and set v := λ(0) ∈ Rn . Then, λi (t) = Piv (γ (t)) for any t ∈ [0, 1] .

216 Davide Vittone

4.4 Extremals in Carnot groups Theorem 4.11 is our main result from a technical viewpoint. Its consequences, however, are probably even more interesting; let us start by discussing its implications in the case of normal extremals. Theorem 4.12. (Characterization of normal extremals in Carnot groups). Let γ : [0, 1] → G be an horizontal curve with γ (0) = 0. Then, the following conditions are equivalent: (a) γ is a normal extremal;  (b) there exists v ∈ Rn such that γ˙ = − ri=1 Piv (γ )X i (γ ). In particular, the sum P1v (γ )2 + · · · + Prv (γ )2 is constant on [0, 1]. Proof. Let us begin with the implication (a)⇒(b). Let h be the controls associated with γ and λ be the dual variable; set v := λ(0) ∈ Rn . By (4.6) and Theorem 4.11, we have h i = −λi = −Piv (γ ) on [0, 1],

∀i = 1, . . . , n ,

and (b) immediately follows. The fact that P1v (γ )2 + · · · + Prv (γ )2 is constant is equivalent to |h| being constant. Concerning the implication (b)⇒(a), notice that the controls h i = −Piv (γ ), together with the functions λi := Piv (γ ), satisfy (4.6) (with ξ0 = 1) and (4.7), because r d  v  (b)  λ˙ i = P jv (γ ) X j Piv (γ ) Pi (γ ) = − dt j=1 r n (4.13)  

r  n 

j=1 k=1

j=1 k=1

(−P jv (γ ))ckji Pkv (γ ) = −

=

cikj h j λk .

This proves that γ is a normal extremal with dual curve λ, as desired. Theorem 4.12 characterizes normal extremals as solutions to a certain ODE: notice that we have reduced the 2n-variables Hamiltonian system of Exercise 3.10 to a system of ODEs in n variables. Recalling that left-invariant vector Velds are analytic, by Theorem 4.12 one can improve Proposition 3.12 on the regularity of normal extremals. Corollary 4.13. Let γ : [0, 1] → G be a normal extremal; then, γ is analytic regular.

217 The regularity problem for sub-Riemannian geodesics

Let us now examine the case of abnormal extremals. If λ is the dual curve associated with an abnormal extremal γ , then (4.6) and Theorem 4.11 imply that λi = Piv (γ ) = 0 on [0, 1]

∀i = 1, . . . , r,

provided v := λ(0) ∈ Rn . Moreover, by (4.4), (4.5) and the fact that the basis X 1 , . . . , X n is adapted to the stratiVcation, the Goh condition (3.10) is equivalent to λi = Piv (γ ) = 0 on [0, 1] for any i = r + 1, . . . , r2 . Recall that the integer r2 has been deVned as dim V1 + dim V2 . We have therefore Theorem 4.14. (Characterization of abnormal extremals in Carnot groups). Let γ : [0, 1] → G be an horizontal curve with γ (0) = 0. Then, the following conditions are equivalent: (a) γ is an abnormal extremal; (b) there exists v ∈ Rn \ {0} such that P1v (γ ) = · · · = Prv (γ ) = 0. v Moreover, the Goh condition (3.10) holds if and only if Pr+1 (γ ) = · · · = v Pr2 (γ ) = 0.

The proof is left as an exercise to the reader, who will also notice that the parameter v ∈ Rn is equal to λ(0), which is not zero due to Theorem 3.6 (i). Remark 4.15. The fact that v = 0 implies that there exist at least one index i ∈ {1, . . . , r} and another index j ∈ {r + 1, . . . , r2 } such that neither Piv nor P jv are the null polynomial; see [18, Proposition 2.6] for more details. In particular, any abnormal extremal γ belongs to an algebraic variety (the one deVned by the equalities in Theorem 4.14 (b)) that is not trivial. The characterization of abnormal extremals in Carnot groups allows for several applications; here, we are going to recall a few of those presented in [17] and [18]. It is possible to construct very irregular abnormal extremals satisfying also the Goh condition. For instance, there exists a 32-dimensional Carnot group G such that, for any Lipschitz function ϕ : [0, 1] → G, there exists a Goh abnormal extremal of the form γ (t) = (t 2 , t, ϕ(t), ∗, . . . , ∗) .

218 Davide Vittone

See [17, Section 6.4] for more details. In the same spirit, a “spiral-like” abnormal Goh extremal has been provided in [18, Section 5]. These examples somehow suggest that a Vner analysis of necessary conditions is needed if one aims at proving smoothness of minimizers, since even second-order necessary conditions (the Goh one) are not enough to ensure regularity. W. Liu and H. J. Sussman have proved in [23] that, if γ is an abnormal extremal in a CC structure of rank r = 2 with dual curve ξ satisfying ξ(t)  ⊥ [[, ], ]γ (t)

for any t ∈ [0, 1] ,

then γ is smooth. Abnormal extremals satisfying the previous condition are called regular abnormal and are somehow “generic”; let us recall that the Goh condition ξ ⊥ ( ∪ [, ])γ holds for abnormal extremals because of Remark 3.22. When working in Carnot groups of rank 2, the regularity of such extremals can be proved in a plain way by using extremal polynomials, see [17, Section 6.2]. The results in [23] are anyway much Vner, as they show (in a more general framework) that regular abnormal extremals are also locally minimizing. In the following sections we analyze with more details two further applications of our machinery.

5 Minimizers in step 3 Carnot group In this section, we review the proof given in [17, Section 6.1] of the following result, that was Vrst proved by K. Tan and X. Yang in [38]. Theorem 5.1. Any minimizer in a Carnot group of step 3 is C ∞ smooth. Proof. By contradiction, assume that there exists a length minimizing curve γ : [0, 1] → G that is not of class C ∞ ; then, γ is a strictly abnormal minimizer and satisVes the Goh condition. By left invariance, we can assume that γ (0) = 0. By Theorem 4.14 and Remark 4.15, there exist v ∈ Rn \ {0} and j ∈ {r + 1, . . . , r2 } such that P jv is not the null polynomial and (5.1) P jv (γ ) = 0 on [0, 1]. By Exercise 4.9, P jv has homogeneous degree at most 1, hence there exists (a1 , . . . , ar ) ∈ Rr \ {0} such that P jv (x) = a1 x1 + · · · + ar xr ,

(5.2)

where we have also used the fact that P jv (0) = 0. DeVne the left-invariant horizontal vector Veld Y1 := a1 X 1 + · · · + ar X r and complete it to a basis

219 The regularity problem for sub-Riemannian geodesics

Y1 , . . . , Yr of V1 . Using (4.3), (5.1) and (5.2), we obtain that γ˙ is of the form γ˙ = h 2 Y2 (γ ) + · · · + h r Yr (γ ) . Hence, γ is contained in the subgroup of G associated with the Lie subalgebra of g generated by Y2 , . . . , Yr and, in particular, it is contained in a Carnot group of rank r − 1 and step (at most) 3. An easy argument by induction on the rank of the group allows to conclude.

6 On the negligibility of the abnormal set In this Section we review the results contained in [18, Section 4]; to this end, we have to introduce some preliminary notions. The Tanaka prolongation Prol g of a stratiVed Lie algebra g = V1 ⊕ · · · ⊕ Vs is the largest stratiVed Lie algebra which can be written in the form Prol g = · · · ⊕ V−1 ⊕ V0 ⊕ V1 ⊕ · · · ⊕ Vs and with the property that [Vi , V j ] ⊂ Vi+ j for any i  s, j  s. Here, “largest” means that any other extension of g with these properties is (isomorphic to) a sub-algebra of Prol g. The explicit construction of Prol g was provided by N. Tanaka in [39]. The prolongation is never trivial, in the sense that Prol g = g; indeed, it can be proved that dim V0  1. Notice that the number of layers in Prol g in not necessarily Vnite; when Prol g is inVnite dimensional we say that G is nonrigid. Let X 1 , . . . , X n be an adapted basis of g; let us extend it to an adapted basis of Prol g . . . . . . , X − j , . . ., . . . , X −1 , X 0 , X 1 , . . . , X r , . . . . . . , . . . , X n . 5 67 8 67 8 5 67 8 67 8 5 5 basis of V−1

basis of V0

basis of V1

basis of Vs

With a slight abuse of notation, we denote this basis by (X i )i n , where the notation “i  n” means • either −∞ < i  n, if dim Prol g = ∞; • or −m  i  n, if m ∈ N is such that dim Prol g = m + n + 1. We will adopt a similar convention for notations like “i  r” and “i  0”. The Lie algebra Prol g possesses its own structure constants and generalized structure constants. We still denote these constants (that are dek because they clearly Vned for i, j, k  n and α ∈ Nn ) by cikj and ciα coincide with those of g when 1  i, j, k  n. Hence, as in DeVnition

220 Davide Vittone

4.8, for any Vxed v ∈ Rn and any i  n (i.e., also for i  0) one can deVne the extremal polynomial Piv (x) :=

n  (−1)|α| α∈Nn k=1

α!

k vk x α , ciα

x ∈ G,

(6.1)

where we agree that vk = 0 whenever k  0. As in Theorem 4.10, one can prove that Piv (0) = vi for any i  n  X j Piv = ckji Pkv for any i  n, 1  j  n.

(6.2)

k n

These formulae are key tools in the proof of the following result; see [18] for more details. Theorem 6.1. Let γ : [0, 1] → G be an abnormal extremal with γ (0) = 0. Then, there exists v ∈ Rn such that Piv (γ ) = 0 on [0, 1]

for any i  r .

(6.3)

If the Goh condition holds, then the previous formula holds for any i  r2 . Proof. Given the formulae (6.2), the proof is quite elementary and similar to that of Lemma 4.7. We prove (6.3) by reverse induction on the homogeneous degree1 d(i) of i. We set again v := λ(0), λ being the dual curve associated with γ . The base of the induction is the case d(i) = 1, where (6.3) holds by Theorem 4.14. Assume then that Pkv (γ ) ≡ 0 for any k such that d(i) < d(k)  1. Let h ∈ L ∞ ([0, 1], Rr ) be the controls associated with γ , so that γ˙ =  r j=1 h j X j (γ ). Then r r  d v (6.2)   h j X j Piv (γ ) = h j ckji Pkv (γ ) Pi ◦ γ = dt j=1 j=1 k n

=

r 



j=1

k n d(k)=d(i)+1

h j ckji Pkv (γ ) = 0 ,

i.e., Piv (γ ) is constant and equal to Piv (γ (0)) = 0.

1 Clearly, the homogeneous degree is deVned by d(i) = k ⇔ X ∈ V also for i  0. i k

221 The regularity problem for sub-Riemannian geodesics

Theorem 4.14 states that abnormal extremals in Carnot groups are contained in certain algebraic varieties (of a very speciVc type). Theorem 6.1 improves it because it states that these algebraic varieties can be made smaller, as there are more polynomials (than in Theorem 4.14) that vanish along γ . We show an application of our techniques to the Morse-Sard problem for abnormal extremals. In our opinion, the strategy we follow has chances to be adapted to many Carnot groups; however, we present it only in a speciVc group. Let us consider the free2 Carnot group G of rank 2 and step 4, i.e., the group associated with the stratiVed Lie algebra g = V1 ⊕ V2 ⊕ V3 ⊕ V4 with V1 = span{X 1 , X 2 }, V3 = span{X 4 , X 5 },

V2 = span{X 3 }, V4 = span{X 6 , X 7 , X 8 }

and commutation relations [X 2 , X 1 ] = X 3 [X 3 , X 1 ] = X 4 , [X 3 , X 2 ] = X 5 [X 4 , X 1 ] = X 6 , [X 4 , X 2 ] = [X 5 , X 1 ] = X 7 , [X 5 , X 2 ] = X 8 . Using exponential coordinates of the second type (see [12]), G can be identiVed with R8 in such a way that X 1 = ∂1 x12 x3 x 2 x2 x1 x22 ∂4 + x1 x2 ∂5 − 1 ∂6 − 1 ∂7 − ∂8 . 2 6 2 2 We are going to prove the following result. X 2 = ∂2 − x1 ∂3 +

Theorem 6.2. Let G ≡ R8 be the free Carnot group of rank 2 and step 4. Then, there exists a non-zero polynomial in 8 variables Q : R8 → R such that the following holds: if p ∈ G is the endpoint of an abnormal extremal starting from 0, then Q( p) = 0. In particular, the set of points in G that can be connected to the origin with abnormal extremals is contained in the algebraic variety {x ∈ R8 : Q(x) = 0} and has measure zero.

2 Free means, roughly speaking, that it is the Carnot group with largest dimension among those with rank 4 and step 2; equivalently, that any other such group is (isomorphic to) a quotient of the free one.

222 Davide Vittone

Remark 6.3. Theorems 4.14 and 6.1 show that any abnormal extremal is contained in an algebraic variety whose deVnition depends on a parameter v, i.e., on the extremal itself. On the contrary, by Theorem 6.2 there exists a universal algebraic variety containing all abnormal extremals. Proof. As proved in [41], the Tanaka prolongation of G is of the form Prol g = V0 ⊕ g with dim V0 = 4. Let us extend X 1 , . . . , X 8 to an adapted basis {X i }−3i 8 of Prol g. By Theorem 6.1 we know that, for any abnormal extremal γ : [0, 1] → G with γ (0) = 0, there exists v ∈ R8 such that Piv (γ ) = 0 on [0, 1] for any i = −3, . . . , 3 .

(6.4)

We have also used Remark 3.22, i.e., the fact that the Goh condition holds. In particular, vi = Piv (0) = 0

for i = 1, 2, 3.

Therefore, recalling (6.1), any Piv can be written in the form Piv (x) =

8 

vk Q ik (x),

i = −3, . . . , 8

k=4

for suitable polynomials Q ik (x) that are independent from v. For any i = −3, . . . , 3, let us deVne the map Q i : G → R5 by   Q i (x) = Q i4 (x), Q i5 (x), Q i6 (x), Q i7 (x), Q i8 (x) , so that Piv (x) = (v4 , . . . , v8 ), Q i (x). Hence, (6.4) can be rewritten as Q i (γ (t)) ⊥ (v4 , . . . , v8 ) ∀t ∈ [0, 1], ∀ i = −3, . . . , 3 . In particular, for any t ∈ [0, 1], the seven 5-dimensional vectors Q i (γ (t)), −3  i  3, belong to the vector space (v4 , . . . , v8 )⊥ ⊂ R5 ; this vector space has dimension 4 because (v4 , . . . , v8 ) = 0 due to Theorem 3.6 (i). Hence, any 5 of these 7 vectors are linearly dependent, i.e., any 5 × 5 minor of the 5 × 7 matrix   (6.5) (Q ik (x))−3i 3 = col Q −3 |Q −2 | · · · |Q 3 (x) 4k 8

has determinant 0 at any point x on γ . In particular, the determinant of the minor   col Q −1 |Q 0 |Q 1 |Q 2 |Q 3 (x) is a polynomial Q(x) (independent from v) which vanish along γ . It is now a boring task to prove that Q is not the null polynomial; we refer to the proof of [18, Theorem 4.1] for details. This concludes the proof.

223 The regularity problem for sub-Riemannian geodesics

Remark 6.4. The determinant of any 5 × 5 minor of the matrix in (6.5) has to vanish   along abnormal extremals; hence, in principle, one could produce 75 = 21 polynomials as in the statement of Theorem 6.2. See [18, Remark 4.2.] for a more detailed discussion on these and other considerations.

ACKNOWLEDGEMENTS . It is a pleasure to thank R. Monti for many invaluable comments, remarks and suggestions. We have to thank G. P. Leonardi for suggesting the characterization of normal extremals in Carnot groups contained in Theorem 4.12. We are indebted with E. Le Donne and E. Pasqualetto for their careful reading of a preliminary version of these notes. Finally, we want to thank the organizers G. Alberti, L. Ambrosio and C. De Lellis, as well as all the participants, for the nice time and the pleasant atmosphere during the ERC School on Geometric Measure Theory and Real Analysis.

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CRM Series Publications by the Ennio De Giorgi Mathematical Research Center Pisa

The Ennio De Giorgi Mathematical Research Center in Pisa, Italy, was established in 2001 and organizes research periods focusing on speciVc Velds of current interest, including pure mathematics as well as applications in the natural and social sciences like physics, biology, Vnance and economics. The CRM series publishes volumes originating from these research periods, thus advancing particular areas of mathematics and their application to problems in the industrial and technological arena.

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