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Morse index, Betti numbers and singular set of bounded area minimal hypersurfaces

Song, Antoine (2019) Morse index, Betti numbers and singular set of bounded area minimal hypersurfaces. . (Unpublished)

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We introduce a combinatorial argument to study closed minimal hypersurfaces of bounded area and high Morse index. Let (Mⁿ⁺¹,g) be a closed Riemannian manifold and Σ subset M$ be a closed embedded minimal hypersurface with area at most A > 0 and with a singular set of Hausdorff dimension at most n - 7. We show the following bounds: there is C_A > 0 depending only on n, g, and A so that Σᵢ₌₀ⁿ bᶦ(Σ) ≤ C_A (1 + index(Σ)) if 3 ≤ n + 1 ≤ 7, Hⁿ⁻⁷(Sing(Σ)) ≤ C_A (1 + index(Σ))^(7/n) if n + 1 ≥ 8, where bᶦ denote the Betti numbers over any field, Hⁿ⁻⁷ is the (n - 7)-dimensional Hausdorff measure and Sing(Σ) is the singular set of Σ. In fact in dimension n + 1 = 3, C_A depends linearly on A. We list some open problems at the end of the paper.

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Additional Information:I am grateful to Fernando Codá Marques and André Neves for their continued support. This work benefited from extended discussions with Jonathan J. Zhu. I thank Otis Chodosh, Chao Li, Davi Maximo, Brian White, Xin Zhou for interesting conversations, and Hans-Joachim Hein for explaining to me his unpublished work with Aaron Naber on certain constructions of Kähler-Einstein metrics. I also thank Giada Franz and Santiago Cordero Misteli for correcting a mistake in Lemma 25. I am indebted to the reviewers, whose constructive comments and numerous corrections substantially improved the writing of this article. The author was partially supported by NSF-DMS-1509027. This research was partially conducted during the period the author served as a Clay Research Fellow.
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Clay Mathematics InstituteUNSPECIFIED
Record Number:CaltechAUTHORS:20221026-539125000.6
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:117594
Deposited By: George Porter
Deposited On:27 Oct 2022 22:07
Last Modified:27 Oct 2022 22:07

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