Regularity of stable minimal hypersurfaces

Abstract

In [13] various estimates for the length of the second fundamental form of stable minimal hypersurfaces immersed in a complete Riemannian manifold, including pointwise estimates in case dimension n of the hypersurface satisfied n S 5, were obtained.There were no known examples to discount the possibility that such estimates might hold for hypersurfaces of dimension n 5 6. Here we show that, at least for properly imbedded stable minimal hypersurfaces, it is indeed possible to obtain such bounds in dimension 6. Perhaps more interestingly, we obtain a regularity theorem (Theorem 1) which is valid in arbitrary dimensions. Roughly speaking, this theorem says that whenever a stable minimal hypersurface is sufficiently close to a hyperplane, it decomposes into a finite union of graphs, which (relative to suitable coordinate axes) are graphs of functions of small Cz norm. Results of this type are familiar in case the hypersurface M is assumed to be area minimizing, or in case the area of M is assumed u priori to be close to the area of a disk (see [4], [l 11, [I], [3]).In contrast to the method of [13] which relies heavily on the identity for Laplacian of the second fundamental form (see [14]), our method uses only "lower order" computations.Using Theorem 1, together with a modification of the "dimension reducing" argument of Federer [6] and the result of J. Simons ([14], Lemma 6.1.1) concerning nonexistence of stable minimal cones in dimension less than or equal to 6, we are able to prove estimates for the singular set, and a compactness theorem. These results (given precisely in Theorems 2, 3, and Corollary 1) apply in the context of stable hypersurfaces with sufficiently small singular set. (We need to know u priori that the singular set has (n -2)-dimensional Hausdorff measure zero.)Finally, in Section 7, we discuss applications of these results to the question of existence of embedded minimal hypersurfaces in a given compact Riemannian manifold, extending recent work of Pitts [9]. In fact we show that if N is an arbitrary compact oriented Riemannian manifold, then N contains a properly embedded boundaryless minimal hypersurface M such that m-M has s-dimensional Hausdorff measure zero for each non-negative s > n

Existence, Regularity, and Properties of Generalized Apparent Horizons

“…A summary of results related to what we call the C-almost minimizing property here and the class F C of boundaries with this property is given in [E07, Appendix A] with concise references to the geometric measure theory literature. We derive the C 2,α -estimates from results in [SS81], which we appropriate to our context in Appendix A. The robust 'low order approach' to regularity used here via geometric measure theory and the stability based analysis of [SS81] is available also when n ≥ 8 if we accept thin singular sets.…”

“…We derive the C 2,α -estimates from results in [SS81], which we appropriate to our context in Appendix A. The robust 'low order approach' to regularity used here via geometric measure theory and the stability based analysis of [SS81] is available also when n ≥ 8 if we accept thin singular sets. It provides a satisfactory theory for limits of regular embedded horizons in arbitrary dimensions that is friendly towards analysis, see Remark A.3 for details, and compare with the curvature estimates that were obtained in [AM05] by generalizing the iteration method of [SSY75].…”

“…The pointwise inequality (2) implies a stability-type inequality for G n in a standard way: given a test function φ ∈ C 1 c (U × R), multiply (2) by vφ 2 and integrate by parts on G n to obtain we could take any other positive constant less than 1 (n−1) to make the arguments below work; this cut-off however is critical for the use of [SS81], as we discuss in the appendix.…”

Min-max theory, Willmore conjecture and the energy of links

“…The restriction to dimensions n ≤ 6 in the previous theorem is not necessary if we allow Σ to have a singular set of codimension 7, as it was later shown in [26].…”

Rigidity of Stable Marginally Outer Trapped Surfaces in Initial Data Sets

“…we can follow, almost verbatim, the proof of Lemma 1 in [SS81] in order to get the pointwise geometric inequality…”