Rigidity of min-max minimal spheres in three-manifolds

TL;DR

This paper investigates the rigidity properties of min-max minimal spheres in three-manifolds, establishing conditions under which metrics must contain minimal spheres with specific area and index bounds, and providing sharp estimates for the width.

Contribution

It proves new rigidity results for min-max minimal spheres in three-manifolds with scalar and Ricci curvature bounds, including area and index estimates and width bounds.

Findings

Metrics on 3-spheres with scalar curvature ≥ 6 not round have minimal spheres with area < 4π and index ≤ 1

Positive Ricci curvature implies sharp estimates for the width of the manifold

Rigidity results restrict the geometry of three-manifolds under curvature conditions

Abstract

In this paper we consider min-max minimal surfaces in three-manifolds and prove some rigidity results. For instance, we prove that any metric on a 3-sphere which has scalar curvature greater than or equal to 6 and is not round must have an embedded minimal sphere of area strictly smaller than $4\pi$ and index at most one. If the Ricci curvature is positive we also prove sharp estimates for the width.