Existence of embedded minimal tori in three-spheres with positive Ricci curvature

TL;DR

This paper proves the existence of multiple embedded minimal tori in three-spheres with positive Ricci curvature using Morse inequalities and min-max theory, revealing at least 4 or 9 such tori depending on the metric's properties.

Contribution

It establishes the existence of multiple embedded minimal tori in three-spheres with positive Ricci curvature, extending previous results with new multiplicity bounds.

Findings

At least 4 embedded minimal tori exist in general.

At least 9 embedded minimal tori exist if the metric is bumpy.

The proof uses Morse inequalities and a multiplicity one theorem.

Abstract

In this paper, we prove the strong Morse inequalities for the area functional in the space of embedded tori and spheres in the three sphere. As a consequence, we prove that in the three dimensional sphere with positive Ricci curvature, there exist at least 4 distinct embedded minimal tori. Suppose in addition that the metric is bumpy, then the three-sphere contains at least 9 distinct embedded minimal tori. The proof relies on a multiplicity one theorem for the Simon-Smith min-max theory proved by the second author and X. Zhou.