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 $M^n$ be a closed immersed minimal hypersurface in the unit sphere ${\mathbb{𝕊}}^{n+1}$. We establish a special isoperimetric inequality of $M^n$. As an application, if the scalar curvature of $M^n$ is constant, then we get a uniform lower bound independent of $M^n$ 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