An isoperimetric inequality of minimal hypersurfaces in spheres

TL;DR

This paper proves a special isoperimetric inequality for minimal hypersurfaces in spheres, providing bounds related to scalar curvature and connecting Cheeger's constant with nodal sets, advancing geometric analysis in spherical contexts.

Contribution

It establishes a novel isoperimetric inequality for minimal hypersurfaces in spheres and links scalar curvature and nodal set volume to geometric bounds.

Findings

Derived a specific isoperimetric inequality for minimal hypersurfaces in spheres.

Established a uniform lower bound for the inequality when scalar curvature is constant.

Connected Cheeger's constant with the volume of nodal sets of height functions.

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.