Rigidity of area-minimizing free boundary surfaces in mean convex three-manifolds

TL;DR

This paper establishes local and global rigidity results for area-minimizing free boundary surfaces in three-manifolds with mean convex boundary and scalar curvature bounds, extending known geometric splitting theorems.

Contribution

It introduces a local splitting theorem for such manifolds and applies it to prove a global rigidity theorem for area-minimizing free boundary disks.

Findings

Local splitting theorem for three-manifolds with mean convex boundary

Global rigidity theorem for area-minimizing free boundary disks

Rigidity results for Plateau problem solutions in negative scalar curvature

Abstract

We prove a local splitting theorem for three-manifolds with mean convex boundary and scalar curvature bounded from below that contain certain locally area-minimizing free boundary surfaces. Our methods are based on those of Micallef and Moraru. We use this local result to establish a global rigidity theorem for area-minimizing free boundary disks. In the negative scalar curvature case, this global result implies a rigidity theorem for solutions of the Plateau problem with length-minimizing boundary.