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THE SINGULAR STRUCTURE AND REGULARITY OF STATIONARY VARIFOLDS AARON NABER AND DANIELE VALTORTA ABSTRACT. If one considers an integral varifold Im M with bounded mean curvature, and if S k(I) x ⊆ ≡ { ∈ M : no tangent cone at x is k + 1-symmetric is the standard stratification of the singular set, then it is well } known that dim S k k. In complete generality nothing else is known about the singular sets S k(I). In this ≤ paper we prove for a general integral varifold with bounded mean curvature, in particular a stationary varifold, that every stratum S k(I) is k-rectifiable. In fact, we prove for k-a.e. point x S k that there exists a unique ∈ k-plane Vk such that every tangent cone at x is of the form V C for some cone C. n 1 n × In the case of minimizing hypersurfaces I − M we can go further. Indeed, we can show that the singular ⊆ set S (I), which is known to satisfy dim S (I) n 8, is in fact n 8 rectifiable with uniformly finite n 8 measure. ≤ − − − 7 An effective version of this allows us to prove that the second fundamental form A has apriori estimates in Lweak on I, an estimate which is sharp as A is not in L7 for the Simons cone. In fact, we prove the much stronger | | 7 estimate that the regularity scale rI has Lweak-estimates. The above results are in fact just applications of a new class of estimates we prove on the quantitative k k k k + stratifications S ǫ,r and S ǫ S ǫ,0. Roughly, x S ǫ I if no ball Br(x) is ǫ-close to being k 1-symmetric. We k ≡ ∈ ⊆ k n k show that S is k-rectifiable and satisfies the Minkowski estimate Vol(B S ) C r − . The proof requires a new ǫ r ǫ ≤ ǫ L2-subspace approximation theorem for integral varifolds with bounded mean curvature, and a W1,p-Reifenberg type theorem proved by the authors in [NVa]. CONTENTS 1. Introduction 3 1.1. Results for Varifolds with Bounded Mean Curvature 6 1.2. Results for Minimizing Hypersurfaces 7 1.3. Outline of Proofs and Techniques 8 arXiv:1505.03428v4 [math.DG] 28 Jul 2017 2. Preliminaries 12 2.1. Integral Varifolds and Mean Curvature 12 2.2. Minimizing integral currents 13 2.3. Bounded Mean Curvature and Monotonicity 15 2.4. Quantitative 0-Symmetry and Cone Splitting 15 2.5. ǫ-regularity for Minimizing Hypersurfaces 17 2.6. Hausdorff, Minkowski, and packing Content 17 2.7. Examples 19 2.8. The Classical Reifenberg theorem 20 Date: August 1, 2017. The first author has been supported by NSF grant DMS-1406259, the second author has been supported by SNSF grants 149539 and PZ00P2 168006. 1 2 AARON NABER AND DANIELE VALTORTA 3. The W1,p-Reifenberg theorem 21 3.1. Statement of Main W1,p-Reifenberg and rectifiable-Reifenberg Results 21 3.2. Explanatory example 23 4. Technical Constructions toward New Reifenberg Results 24 4.1. Hausdorff distance and subspaces 25 4.2. Distance between L2 best planes 27 2 4.3. Comparison between L and L∞ planes 30 4.4. bi-Lipschitz equivalences 30 4.5. Pointwise Estimates on D 35 5. Proof of theorem 3.3: The Discrete-Reifenberg 36 5.1. First induction: upwards 37 5.2. Rough estimate 37 5.3. Second induction: downwards. Outline of the proof 38 5.4. First steps in the induction 40 5.5. Second induction: details of the construction 41 5.6. Excess set. 42 5.7. Good, bad and final balls 43 5.8. Map and manifold structure. 44 5.9. Properties of the manifolds Ti′ 46 5.10. Volume estimates on the manifold part 46 5.11. Estimates on the excess set 47 5.12. Volume estimates. 48 5.13. Upper estimates for µ. 49 6. L2-Best Approximation theorems for Stationary varifolds 49 6.1. Symmetry and Gradient Bounds 50 6.2. Best L2-Subspace Equations 50 6.3. Proof of theorem 6.1 53 7. The Inductive Covering lemma 54 7.1. Estimating Ur in lemma 7.1 for r > 0 56 7.2. Estimating U0 in lemma 7.1 58 7.3. Technical Constructions for Estimating U+ 59 7.4. Estimating U+ in lemma 7.1 63 8. Proof of Main theorem’s for Integral Varifolds with Bounded Mean Curvature 66 8.1. Proof of theorem 1.3 66 8.2. Proof of theorem 1.4 68 8.3. Proof of theorem 1.5 71 9. Proof of Main theorem’s for Minimizing Hypersurfaces 71 9.1. Proof of theorem 1.6 71 9.2. Proof of theorem 1.8 72 Acknowledgments 72 References 72 THE SINGULAR STRUCTURE AND REGULARITY OF STATIONARY VARIFOLDS 3 1. INTRODUCTION In this paper we will study integral m-varifolds Im with bounded mean curvature, and in particular sta- tionary varifolds and area minimizing currents, see Section 2 for an introduction to the basics. In the case of a varifold I with bounded mean curvature we can consider the stratification of I given by S 0(I) S k(I) S m(I) I , (1.1) ⊆···⊆ ⊆··· ⊆ where the singular sets are defined as S k(I) x M : no tangent cone at x is k + 1-symmetric , (1.2) ≡{ ∈ } see Definition 1.2 for a precise definition and more detailed discussion. An important result of Almgren (see [Alm00, sections 2.25, 2.26]) tells us that in the context of area minimizing currents we have the Hausdorff dimension estimate dim S k(I) k . (1.3) ≤ With some work, this estimate can be carried over to the context of integral varifolds. Unfortunately, essen- tially nothing else is understood about the structure of the singular sets in any generality. In the case where I is a codimension 1 area minimizing current more is understood, at least for the top stratum of the singular n 1 n 8 set. Namely, if I − is an area minimizing hypersurface it has been shown in [Sim95] that S (I) = S − (I) is rectifiable. This result relies on Simons’ work [Sim68], where the author proves that all tangent cones to hypersurfaces in Rn are hyperplanes if n 7. ≤ Moreover, for a minimizing hypersurface, i.e., a minimizing current of codimension 1, one can use the ǫ- regularity theorem of [Fed69],[CN13b] (see theorem 2.11) to show that the whole singular set S (I) coincides n 8 with S − (I). As it is well-known, this is not the case for minimizing currents of codimension higher than one, as singular points in this context may also arise as branching points on the top stratum S m(I) S m 1(I). \ − The first goal of this paper is to give improved regularity results for stratification and associated quanti- tative stratification for varifolds under only a bounded mean curvature assumption. Indeed, we will show for such a varifold that the strata S k(I) are k-rectifiable for every k. In fact, we will show that for k-a.e. k x S (I) there exists a unique k-dimensional subspace V Tx M such that every tangent cone at x is of the ∈ ⊆ form Vk C for some cone C. Note that we are not claiming that the cone factor C is necessarily unique, × just that the Euclidean factor Vk of the cone is. It is a good question as to whether tangent cones need to be unique in general under only the assumption of bounded mean curvature. m n 1 For a varifold I = I − which is an area minimizing hypersurface these results can be improved. To begin with, we have that the top stratum of the singular set S (I) = S n 8(I) is n 8 rectifiable, and that there − − are a priori bounds on its n 8 Hausdorff measure. That is, if the mass µI(B ) Λ is bounded, then we − 2 ≤ have the estimate Hn 8(S (I) B ) C(Λ, B ). In fact, we have the stronger Minkowski estimate − ∩ 1 ≤ 2 8 Vol Br S (I) B , rµI Br S (I) B Cr , (1.4) 1 1 ≤ ∩ ∩ where Br S (I) is the tube of radius r around the singular set. Indeed, we can prove much more effective versions of these estimates. That is, in theorem 1.8 we show that the second fundamental form A of I, and 4 AARON NABER AND DANIELE VALTORTA 7 in fact the regularity scale rI, have a priori bounds in Lweak. More precisely, we have that 1 1 7 µI A > r− B1 µI Br A > r− B1 C(Λ, B2)r . (1.5) n| | o ∩ ≤ n| | o ∩ ≤ Let us observe that these estimates are sharp, in that the Simons’ cone satisfies A < L7 (I). Let us also | | loc point out that this sharpens estimates of [CN13b], where it was proven that for minimizing hypersurfaces that A Lp for all p < 7. We refer the reader to Section 1.2 for the precise and most general statements. | |∈ Now the techniques of this paper all center around the notion of the quantitative stratification. In fact, it is for the quantitative stratification that the most important results of the paper hold, everything else can be seen to be corollaries of these statements. The quantitative stratification was first introduced in [CN13a], and later used in [CN13b] with the goal of giving effective and Lp estimates on stationary and minimizing currents. It has since been used in [CHN13b], [CHN13a], [CNV15], [FMS15], [BL15] to prove similar results in the areas of mean curvature flow, critical sets of elliptic equations, harmonic map flow.
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  • Max-Planck-Institut Für Mathematik in Den Naturwissenschaften Leipzig

    Max-Planck-Institut Für Mathematik in Den Naturwissenschaften Leipzig

    Max-Planck-Institut fur¨ Mathematik in den Naturwissenschaften Leipzig Rectifiability of varifolds with locally bounded first variation with respect to anisotropic surface energies by Guido De Philippis, Antonio De Rosa, and Francesco Ghiraldin Preprint no.: 61 2016 RECTIFIABILITY OF VARIFOLDS WITH LOCALLY BOUNDED FIRST VARIATION WITH RESPECT TO ANISOTROPIC SURFACE ENERGIES GUIDO DE PHILIPPIS, ANTONIO DE ROSA, AND FRANCESCO GHIRALDIN Abstract. We extend Allard's celebrated rectifiability theorem to the setting of varifolds with locally bounded first variation with respect to an anisotropic integrand. In particular, we identify a sufficient and necessary condition on the integrand to obtain the rectifiability of every d-dimensional varifold with locally bounded first variation and positive d-dimensional density. In codimension one, this condition is shown to be equivalent to the strict convexity of the integrand with respect to the tangent plane. 1. Introduction n Allard's rectifiability Theorem, [1], asserts that every d-varifold in R with locally bounded (isotropic) first variation is d-rectifiable when restricted to the set of points n in R with positive lower d-dimensional density. It is a natural question whether this result holds for varifolds whose first variation with respect to an anisotropic integrand is locally bounded. n 1 More specifically, for an open set Ω ⊂ R and a positive C integrand F :Ω × G(n; d) ! R>0 := (0; +1); n where G(n; d) denotes the Grassmannian of d-planes in R , we define the anisotropic energy of a d-varifold V 2 Vd(Ω) as Z F(V; Ω) := F (x; T ) dV (x; T ): (1.1) Ω×G(n;d) We also define its anisotropic first variation as the order one distribution whose 1 n action on g 2 Cc (Ω; R ) is given by d # δF V (g) := F 't V dt t=0 Z h i = hdxF (x; T ); g(x)i + BF (x; T ): Dg(x) dV (x; T ); Ω×G(n;d) # n where 't(x) = x+tg(x), 't V is the image varifold of V through 't, BF (x; T ) 2 R ⊗ n ∗ n n R is an explicitly computable n×n matrix and hA; Bi := tr A B for A; B 2 R ⊗R , see Section 2 below for the precise definitions and the relevant computations.