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

Understand

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.