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

We prove splitting theorems for mean convex open subsets in Riemannian curvature-dimension (RCD) spaces that extend results by Kasue et al. 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 with respect 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.