Existence of four minimal spheres in $S^3$ with a bumpy metric

TL;DR

This paper proves that in the 3-sphere with certain curved metrics, there are at least four distinct minimal spheres, confirming a long-standing conjecture and using advanced min-max theory techniques.

Contribution

It establishes the existence of at least four minimal spheres in the 3-sphere with bumpy or positive Ricci curvature metrics, confirming Yau's conjecture from 1982.

Findings

Existence of at least four minimal spheres in the specified metrics

Confirmation of Yau's 1982 conjecture for these metrics

Application of a multiplicity one theorem in min-max theory

Abstract

We prove that in the three dimensional sphere with a bumpy metric or a metric with positive Ricci curvature, there exist at least four distinct embedded minimal two-spheres. This confirms a conjecture of S. T. Yau in 1982 for bumpy metrics and metrics with positive Ricci curvature. The proof relies on a multiplicity one theorem for the Simon-Smith min-max theory.