Stable minimal hypersurfaces in $\mathbf{R}^4$

Abstract

More generally, we recall that a minimal immersion M 3 !(N 4 , g) is stable iffor all f C 0 (M \M ).Theorem 1.3.Let (N 4 , g) be a closed Riemannian manifold.There exists C = C(N, g) such that, if M 3 !N 4 is a 2-sided, stable minimal immersion, thenWe have recently generalized Theorem 1.3 to hold in (non-compact) ambient (N 4 , g) with bounded sectional curvature in a joint work with Stryker [18, Corollary 2.5], which resolves [22, Conjecture 2.13].Theorems 1.2 and 1.3 are the 4-dimensional analogue of the well-known curvature estimate of Schoen [46] for minimal surfaces in three dimensions.Note that, by the work of Schoen-Simon-Yau [49], such an estimate was previously known to hold where C depended on an upper bound for volume of M 3 in small balls.Remark 1.4.There have been several interesting developments since the first version of this paper was posted.The authors have discovered [16] a new proof of Theorem 1.1 that can be localized to obtain (interior) volume estimates (in the spirit of Pogorelov [45]; cf.[20]).This new proof is related to the study of uniformly positive scalar curvature, while the current paper is related to the study of non-negative scalar curvature.Subsequently, Catino-Mastrolia-Roncoroni have discovered [11] a completely different proof of Theorem 1.1, related to the study of Bakry-mery-Ricci curvature.Interestingly, the dimension restriction n+1=4 enters each proof in a different way.( 1 ) A slight modification of the proof of Theorem 1.1 yields a structure theorem for finite-index minimal hypersurfaces in R 4 , analogous to the well-known results of Gulliver, Fischer-Colbrie, and Osserman [27], [30], [43].Recall that a complete, 2-sided, immersedwhere( 1 ) Added in proof: Theorem 1.1 has recently been generalized to the cases M 4 !R 5 by the authors along with Minter and Stryker [17], as well as M 5 !R 6 by Mazet [36].Nktp4jcrcuP09pwXQggfOTSZkVxS7gfwjwEzfN3Ci1c= Nktp4jcrcuP09pwXQggfOTSZkVxS7gfwjwEzfN3Ci1c= stable minimal hypersurfaces in R 4 3 Theorem 1.5.A complete, 2-sided, minimal immersion M 3 !R 4 has finite Morse index if and only if it has finite total curvature M |A M | 3 <.We remark that Tysk [59] proved the same statement for a complete, 2-sided, minimal immersion M n !R n+1 (for 3n6) under the assumption that M has Euclidean volume growth.Theorem 1.5 has strong consequences on the structure of M near infinity.We recall the following definition.Definition 1.6.([47, 2]) Suppose n3 and let M n !R n+1 be a complete minimal immersion.An end E of M is regular at infinity if it is the graph of a function w over a hyperplane with the asymptoticsfor some constants a, b, and c j , where x 1 , ..., x n are the coordinates in .mersion with finite total curvature, then each end of M is regular at infinity.Moreover, by [34], a 2-sided minimal immersion with finite Morse index has finitely many ends.Combined with Theorem 1.5, this yields the following result.Corollary 1.7.Suppose M 3 !R 4 is a complete, 2-sided, minimal immersion with finite Morse index.Then M has finitely many ends, each of which is regular at infinity.In particular, M has cubic volume growth, i.e. sup

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