Minimal two-spheres in three-spheres

TL;DR

This paper proves that generic three-spheres contain at least two embedded minimal two-spheres, using min-max theory and mean curvature flow, and applies these results to solve longstanding problems in geometric analysis.

Contribution

It establishes the existence of multiple minimal two-spheres in generic three-spheres and introduces new techniques combining min-max theory with mean curvature flow.

Findings

Existence of at least two embedded minimal two-spheres in generic three-spheres.

Construction of smooth mean convex foliations in three-manifolds.

Resolution of Yau's 1987 problem on minimal spheres in ellipsoids.

Abstract

We prove that any manifold diffeomorphic to $S^3$ and endowed with a generic metric contains at least two embedded minimal two-spheres. The existence of at least one minimal two-sphere was obtained by Simon-Smith in 1983. Our approach combines ideas from min-max theory and mean curvature flow. We also establish the existence of smooth mean convex foliations in three manifolds. We apply our methods to solve a problem posed by S.T. Yau in 1987 on whether the planar two-spheres are the only minimal spheres in ellipsoids centered about the origin in $\mathbb{R}^4$. Finally, considering the example of degenerating ellipsoids we show that the assumptions in the multiplicity one conjecture and the equidistribution of widths conjecture are in a certain sense sharp.