15 August 2023 Morse index, Betti numbers, and singular set of bounded area minimal hypersurfaces
Antoine Song
Author Affiliations +
Duke Math. J. 172(11): 2073-2147 (15 August 2023). DOI: 10.1215/00127094-2023-0012


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 let ΣM be a closed embedded minimal hypersurface with area at most A>0 and with a singular set of Hausdorff dimension at most n7. We show the following bounds: there is CA>0 depending only on n, g, and A so that


where bi denote the Betti numbers over any field, Hn7 is the (n7)-dimensional Hausdorff measure, and Sing(Σ) is the singular set of Σ. In fact, in dimension n+1=3, CA depends linearly on A. We list some open problems at the end of the paper.


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Antoine Song. "Morse index, Betti numbers, and singular set of bounded area minimal hypersurfaces." Duke Math. J. 172 (11) 2073 - 2147, 15 August 2023. https://doi.org/10.1215/00127094-2023-0012


Received: 29 February 2020; Revised: 22 June 2022; Published: 15 August 2023
First available in Project Euclid: 26 June 2023

MathSciNet: MR4627248
zbMATH: 07732803
Digital Object Identifier: 10.1215/00127094-2023-0012

Primary: 53A10

Keywords: Betti numbers , minimal surfaces , Morse index , singular set

Rights: Copyright © 2023 Duke University Press

Vol.172 • No. 11 • 15 August 2023
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