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An isoperimetric inequality of minimal hypersurfaces in spheres

Fagui Li and Niang Chen

Vol. 324 (2023), No. 1, 143–156
DOI: 10.2140/pjm.2023.324.143
Abstract

Let Mn be a closed immersed minimal hypersurface in the unit sphere 𝕊n+1. We establish a special isoperimetric inequality of Mn. As an application, if the scalar curvature of Mn is constant, then we get a uniform lower bound independent of Mn for the isoperimetric inequality. In addition, we obtain an inequality between Cheeger’s isoperimetric constant and the volume of the nodal set of the height function.

Keywords
isoperimetric inequality, minimal hypersurface, nodal set, Cheeger's isoperimetric constant
Mathematical Subject Classification
Primary: 53A10, 53C24, 53C42
Milestones
Received: 13 March 2022
Revised: 24 November 2022
Accepted: 14 April 2023
Published: 22 June 2023
Authors
Fagui Li
Beijing International Center for Mathematical Research
Peking University
Beijing
China
School of Mathematical Sciences
Laboratory of Mathematics and Complex Systems
Beijing Normal University
Beijing
China
Niang Chen
Department of Mathematics
Faculty of Arts and Sciences
Beijing Normal University at Zhuhai
Zhuhai
China

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