Rigidity of mean convex subsets in non-negatively curved RCD spaces and stability of mean curvature bounds

TL;DR

This paper extends classical geometric theorems to non-smooth RCD spaces, establishing splitting results, stability of mean curvature bounds, and almost rigidity theorems for mean convex domains.

Contribution

It introduces splitting theorems and stability results for mean convex subsets in non-smooth RCD spaces, generalizing known Riemannian results to a broader setting.

Findings

Proved splitting theorems for mean convex subsets in RCD spaces.

Established stability of mean curvature bounds under boundary distance convergence.

Derived almost rigidity results for domains with boundary mean curvature bounds.

Abstract

We prove splitting theorems for mean convex open subsets in RCD (Riemannian curvature-dimension) spaces that extend results by Kasue, Croke and Kleiner for Riemannian manifolds with boundary to a non-smooth setting. A corollary is for instance Frankel's theorem. Then, we prove that our notion of mean curvature bounded from below for the boundary of an open subset is stable w.r.t. to uniform convergence of the corresponding boundary distance function. We apply this to prove almost rigidity theorems for uniform domains whose boundary has a lower mean curvature bound.