Hawking mass and local rigidity of minimal two-spheres in three-manifolds

TL;DR

This paper investigates the local rigidity of minimal two-spheres that maximize Hawking mass in three-manifolds with positive scalar curvature, establishing isometric neighborhoods and characterizing critical points on deSitter-Schwarzschild manifolds.

Contribution

It proves local rigidity results for minimal two-spheres maximizing Hawking mass, including isometric neighborhood structures and characterizations on deSitter-Schwarzschild manifolds.

Findings

Neighborhoods of stable minimal two-spheres are isometric to deSitter-Schwarzschild metrics.

Slices in deSitter-Schwarzschild are the only small graph critical points for Hawking mass.

Slices are local maxima of Hawking mass among small graph competitors.

Abstract

We study rigidity of minimal two-spheres $\Sigma$ that locally maximize the Hawking mass on a Riemannian three-manifold with a positive lower bound on its scalar curvature. After assuming strict stability of $\Sigma$, we prove that a neighborhood of it in $M$ is isometric to one of the deSitter-Schwarzschild metrics on $(- \epsilon,\epsilon)\times \Sigma$. We also show that if $\Sigma$ is a critical point for the Hawking mass on the deSitter-Schwarzschild manifold $\mathbb{R}\times\Sph^2$ and can be written as a graph over a slice $\Sigma_r=\{r\}\times\mathbb{S}^2$, then $\Sigma$ itself must be a slice, and moreover that slices are indeed local maxima amongst competitors that are graphs with small $C^2$-norm.