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November 2020 On the Multiplicity One Conjecture in min-max theory
Xin Zhou
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Ann. of Math. (2) 192(3): 767-820 (November 2020). DOI: 10.4007/annals.2020.192.3.3

Abstract

We prove that in a closed manifold of dimension between $3$ and $7$ with a bumpy metric, the min-max minimal hypersurfaces associated with the volume spectrum introduced by Gromov, Guth, Marques-Neves, are two-sided and have multiplicity one. This confirms a conjecture by Marques-Neves.

We prove that in a bumpy metric each volume spectrum is realized by the min-max value of certain relative homotopy class of sweepouts of boundaries of Caccioppoli sets. The main result follows by approximating such min-max value using the min-max theory for hypersurfaces with prescribed mean curvature established by the author with Zhu.

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Xin Zhou. "On the Multiplicity One Conjecture in min-max theory." Ann. of Math. (2) 192 (3) 767 - 820, November 2020. https://doi.org/10.4007/annals.2020.192.3.3

Information

Published: November 2020
First available in Project Euclid: 21 December 2021

Digital Object Identifier: 10.4007/annals.2020.192.3.3

Subjects:
Primary: 49J35 , 49Q05 , 53C42 , 58E12

Keywords: hypersurfaces with prescribed mean curvature , Minimal hypersurfaces , min-max theory , multiplicity , volume spectrum

Rights: Copyright © 2020 Department of Mathematics, Princeton University

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Vol.192 • No. 3 • November 2020
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