Rigidity of free boundary minimal disks in mean convex three-manifolds

TL;DR

This paper investigates the rigidity properties of free boundary minimal disks in mean convex three-manifolds, showing under certain stability conditions that neighborhoods are isometric to half de Sitter-Schwarzschild space.

Contribution

It establishes a rigidity result for free boundary minimal disks that locally maximize the modified Hawking mass, linking stability to geometric isometry with a model space.

Findings

Neighborhoods of strictly stable free boundary minimal disks are isometric to half de Sitter-Schwarzschild space.

The result applies to manifolds with positive scalar curvature and mean convex boundary.

Provides a rigidity characterization under stability assumptions.

Abstract

The purpose of this article is study rigidity of free boundary minimal two-disks that locally maximize the modified Hawking mass on a Riemannian three-manifold with positive lower bound on its scalar curvature and mean convex boundary. Assuming the strict stability of {\Sigma}, we prove that a neighborhood of it in M is isometric to one of the half de Sitter-Schwarzschild space.