An isoperimetric inequality of minimal hypersurfaces in spheres

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.

arXiv comment13 pages, any comments are welcome