Modified Hawking mass and rigidity of three-manifolds with boundary

TL;DR

This paper proves a rigidity theorem for three-dimensional Riemannian manifolds with boundary, showing they are locally isometric to the half anti-de Sitter-Schwarzschild manifold under certain conditions involving a modified Hawking mass.

Contribution

It introduces a new rigidity result linking free boundary minimal disks with the geometry of the ambient manifold, using a modified Hawking mass.

Findings

Area estimates for free boundary strictly stable two-disks.

Manifolds are locally isometric to the half anti-de Sitter-Schwarzschild manifold.

Rigidity holds under conditions of negative scalar curvature and mean convex boundary.

Abstract

In this paper, we prove a rigidity result for three-dimensional Riemannian manifolds with boundary, under the assumption that a free boundary minimal two-disk, which locally maximizes a modified Hawking mass, is embedded in a $3$-dimensional Riemannian manifold with negative scalar curvature and mean convex boundary. First, we establish area estimates for free boundary strictly stable two-disks. Finally, we show that the $3$-dimensional Riemannian manifold with boundary is locally isometric to the half anti-de Sitter-Schwarzschild manifold.