Annals of Mathematics 179 (2014), 843{1007 http://dx.doi.org/10.4007/annals.2014.179.3.2 A general regularity theory for stable codimension 1 integral varifolds By Neshan Wickramasekera Abstract We give a necessary and sufficient geometric structural condition, which we call the α-Structural Hypothesis, for a stable codimension 1 integral varifold on a smooth Riemannian manifold to correspond to an embedded smooth hypersurface away from a small set of generally unavoidable sin- gularities. The α-Structural Hypothesis says that no point of the support of the varifold has a neighborhood in which the support is the union of three or more embedded C1,α hypersurfaces-with-boundary meeting (only) along their common boundary. We establish that whenever a stable integral n-varifold on a smooth (n + 1)-dimensional Riemannian manifold satisfies the α-Structural Hypothesis for some α 2 (0; 1=2), its singular set is empty if n ≤ 6, discrete if n = 7 and has Hausdorff dimension ≤ n − 7 if n ≥ 8; in view of well-known examples, this is the best possible general dimension estimate on the singular set of a varifold satisfying our hypotheses. We also establish compactness of mass-bounded subsets of the class of stable codimension 1 integral varifolds satisfying the α-Structural Hypothesis for some α 2 (0; 1=2). The α-Structural Hypothesis on an n-varifold for any α 2 (0; 1=2) is readily implied by either of the following two hypotheses: (i) the vari- fold corresponds to an absolutely area minimizing rectifiable current with no boundary, (ii) the singular set of the varifold has vanishing (n − 1)- dimensional Hausdorff measure. Thus, our theory subsumes the well-known regularity theory for codimension 1 area minimizing rectifiable currents and settles the long standing question as to which weakest size hypothesis on the singular set of a stable minimal hypersurface guarantees the validity of the above regularity conclusions. An optimal strong maximum principle for stationary codimension 1 in- tegral varifolds follows from our regularity and compactness theorems. Contents 1. Introduction 844 2. Notation 854 3. Statement of the main theorems 855 c 2014 Department of Mathematics, Princeton University. 843 844 NESHAN WICKRAMASEKERA 4. Proper blow-up classes 861 5. Lipschitz approximation and coarse blow-ups 869 6. An outline of the proof of the main theorems 873 7. Nonconcentration of tilt-excess 875 8. Properties of coarse blow-ups: Part I 879 9. Properties of coarse blow-ups: Part II 888 10. Parametric L2-estimates in terms of fine excess 894 11. Blowing up by fine excess 919 12. Continuity estimates for the fine blow-ups and their derivatives 923 13. Improvement of fine excess 936 14. Properties of coarse blow-ups: Part III 947 15. The Sheeting Theorem 957 16. The Minimum Distance Theorem 968 17. The Regularity and Compactness Theorem 994 18. Generalization to Riemannian manifolds 996 19. A sharp varifold maximum principle 1004 References 1005 1. Introduction Here we study regularity properties of stable critical points of the n-dimen- sional area functional in a smooth (n + 1)-dimensional Riemannian manifold, addressing, among a number of other things, the following basic question: When is a stable critical point V of the n-dimensional area functional in a smooth (n + 1)-dimensional Riemannian manifold made-up of pairwise disjoint, smooth, embedded, connected hypersurfaces each of which is itself a critical point of area? Without further hypothesis, V need not satisfy the stated property; this is illustrated by (a sufficiently small region of) any stationary union of three or more hypersurfaces-with-boundary meeting along a common (n−1)-dimensional submanifold (e.g., a pair of transverse hyperplanes in a Euclidean space). In each of these examples, the connected components of the regular part of the union are not individually critical points of area (in the sense of having vanish- ing first variation with respect to area for deformations by compactly supported smooth vector fields of the ambient space; see precise definition in Section 3). We give a geometrically optimal answer to the above question by estab- lishing a precise version (given as Corollary 1) of the following assertion: Presence of a region of V where three or more hypersurfaces-with- boundary meet along their common boundary is the only obstruction for V to correspond to a locally finite union of pairwise disjoint, smooth, STABLE CODIMENSION 1 INTEGRAL VARIFOLDS 845 embedded, connected hypersurfaces each of which is itself a critical point of area. This follows directly from our main theorem (the Regularity and Compact- ness Theorem) which establishes a precise version of the following regularity statement: Presence of a region of V where three or more hypersurfaces-with- boundary meet along their common boundary is the only obstruction to complete regularity of V in low dimensions and to regularity of V away from a small, quantifiable, set of generally unavoidable singular- ities in general dimensions. In proving these results, we shall first work in the context where the ambient manifold is an open subset of Rn+1 with the Euclidean metric. The differences that arise in the proof in replacing Euclidean ambient space by a general smooth (n + 1)-dimensional Riemannian manifold amount to \error terms" in various identities and inequalities that are valid in the Euclidean setting, and they can be handled in a straightforward manner. We shall discuss this further in the penultimate section of the paper. Here, a critical point of the n-dimensional area means a stationary integral n-varifold; i.e., an integral n-varifold having zero first variation with respect to area under deformation by the flow generated by any compactly supported C1 vector field of the ambient space (see hypothesis (S1) in Section 3). For a varifold V , let reg V denote its regular part, i.e., the smoothly embedded part (of the support of the weight measure kV k associated with V ), and let sing V denote its singular set, i.e., the complement of reg V (in the support of kV k); see Section 2 for the precise definitions of these terms. A stationary integral varifold V is stable if reg V is stable in the sense that V has nonnegative second variation with respect to area under deformation by the flow generated by any C1 ambient vector field that is compactly supported away from sing V and that, on reg V , is normal to reg V . In our codimension 1 setting and for Euclidean ambient space, stability of V whenever reg V is orientable is equivalent to requiring that reg V satisfies the following stability inequality ([Sim83, x9]): Z Z 2 2 n 2 n 1 jAj ζ dH ≤ jr ζj dH 8 ζ 2 Cc (reg V ); reg V reg V here A denotes the second fundamental form of reg V , jAj the length of A, r the gradient operator on reg V and Hn is the n-dimensional Hausdorff measure on Rn+1: (In fact a slightly weaker form of the stability hypothesis suffices for the proofs of all of our theorems here, and as a result of that, orientability of reg V for the varifolds V considered here is a conclusion rather than a hypothesis; see hypothesis (S2) in Section 3 and Corollary 3.2.) 846 NESHAN WICKRAMASEKERA By a stable integral n-varifold we mean a stable, stationary integral n-vari- fold. For α 2 (0; 1) and V an integral varifold on a smooth Riemannian manifold, we state the following condition (hypothesis (S3) in Section 3), which we shall refer to often throughout the rest of the introduction: α-Structural Hypothesis. No singular point of V has a neighbor- hood in which V corresponds to a union of embedded C1,α hypersurfaces-with- boundary meeting (only) along their common C1,α boundary (and with mul- tiplicity a constant positive integer on each of the constituent hypersurfaces- with-boundary). Our main theorem (Theorem 18.1; for Euclidean ambient space, Theo- rem 3.1) can now be stated as follows: Regularity and Compactness Theorem. A stable integral n-varifold V on a smooth (n + 1)-dimensional Riemannian manifold corresponds to an embedded hypersurface with no singularities when 1 ≤ n ≤ 6; to one with at most a discrete set of singularities when n = 7; and to one with a closed set of singularities having Hausdorff dimension at most n − 7 when n ≥ 8 (and with multiplicity, in each case, a constant positive integer on each connected com- ponent of the hypersurface), provided V satisfies the α-Structural Hypothesis above for some α 2 (0; 1=2): Furthermore, for any given α 2 (0; 1=2), each mass-bounded subset of the class of stable codimension 1 integral varifolds satisfying the α-Structural Hypothesis is compact in the topology of varifold convergence. In case V corresponds to an absolutely area minimizing codimension 1 rectifiable current, the regularity conclusion of this theorem is well known and is the result of combined work of E. De Giorgi [DG61], R. Reifenberg [Rei60], W. Fleming [Fle62], F. Almgren [Alm66], J. Simons [Sim68] and H. Federer [Fed70]. While our work uses ideas and results from some of these pioneering works, it does not rely upon the fact that the conclusions hold in the area min- imizing case; it is interesting to note that the above theorem indeed subsumes the regularity theory for codimension 1 area minimizing rectifiable currents for the following simple reason: If T is a rectifiable current on an open ball and if T has no boundary in the interior of the ball and is supported on a union of three or more embedded hypersurfaces-with-boundary meeting only along their common boundary, then T cannot be area minimizing.