The size of isoperimetric surfaces in 3-manifolds and a rigidity result for the upper hemisphere

TL;DR

This paper characterizes the standard 3-sphere as the unique Ricci-positive 3-manifold with scalar curvature at least 6 that maximizes isoperimetric surface area, and proves a rigidity result for the upper hemisphere.

Contribution

It provides a characterization of the standard 3-sphere via isoperimetric surface areas and confirms a special case of Min-Oo's scalar curvature rigidity conjecture for the upper hemisphere.

Findings

Standard 3-sphere uniquely maximizes isoperimetric surface area among Ricci-positive 3-manifolds.

Proves a rigidity result for the upper hemisphere with scalar curvature at least 6.

Answers a special case of Min-Oo's conjecture affirmatively.

Abstract

We characterize the standard $\mathbb{S}^3$ as the closed Ricci-positive 3-manifold with scalar curvature at least 6 having isoperimetric surfaces of largest area: $4\pi$. As a corollary we answer in the affirmative an interesting special case of a conjecture of Min-Oo's on the scalar curvature rigidity of the upper hemisphere..