Existence of infinitely many minimal hypersurfaces in closed manifolds

TL;DR

This paper proves that in closed Riemannian manifolds of dimension 3 to 7, there are infinitely many minimal hypersurfaces, confirming a conjecture by S.-T. Yau using min-max theory.

Contribution

It establishes the existence of infinitely many minimal hypersurfaces in certain manifolds, advancing the understanding of geometric structures.

Findings

Infinitely many minimal hypersurfaces exist in closed manifolds of dimension 3 to 7.

The proof confirms Yau's conjecture on minimal hypersurfaces.

Builds on and extends methods by Marques and Neves.

Abstract

Using min-max theory, we show that in any closed Riemannian manifold of dimension at least 3 and at most 7, there exist infinitely many smoothly embedded closed minimal hypersurfaces. It proves a conjecture of S.-T. Yau. This paper builds on the methods developed by F. C. Marques and A. Neves.