Area of minimal hypersurfaces

TL;DR

This paper proves Yau's conjecture for minimal rotational hypersurfaces in spheres, showing their areas are either minimal, equal to a specific Clifford hypersurface, or significantly larger, and applies this to estimate entropies of certain self-shrinkers.

Contribution

It confirms Yau's conjecture for a class of minimal hypersurfaces and provides area bounds, with applications to entropy estimates of self-shrinkers.

Findings

Yau's conjecture holds for minimal rotational hypersurfaces.

The area of such hypersurfaces is either minimal, a specific Clifford hypersurface, or exceeds a certain threshold.

Entropy estimates are obtained for some special self-shrinkers.

Abstract

A well-known conjecture of Yau states that the area of one of Clifford minimal hypersurfaces $S^k\big{(}\sqrt{\frac{k}{n}}\, \big{)}\times S^{n-k}\big{(}\sqrt{\frac{n-k}{n}}\, \big{)}$ gives the lowest value of area among all non-totally geodesic compact minimal hypersurfaces in the unit sphere $S^{n+1}(1)$. The present paper shows that Yau conjecture is true for minimal rotational hypersurfaces, more precisely, the area $|M^n|$ of compact minimal rotational hypersurface $M^n$ is either equal to $|S^n(1)|$, or equal to $|S^1(\sqrt{\frac{1}{n}})\times S^{n-1}(\sqrt{\frac{n-1}{n}})|$, or greater than $2(1-\frac{1}{\pi})|S^1(\sqrt{\frac{1}{n}})\times S^{n-1}(\sqrt{\frac{n-1}{n}})|$. As the application, the entropies of some special self-shrinkers are estimated.