Boundary regularity and stability for spaces with Ricci bounded below

Abstract

This paper studies the structure and stability of boundaries in noncollapsed $${{\,\mathrm{RCD}\,}}(K,N)$$ RCD ( K , N ) spaces, that is, metric-measure spaces $$(X,{\mathsf {d}},{\mathscr {H}}^N)$$ ( X , d , H N ) with Ricci curvature bounded below. Our main structural result is that the boundary $$\partial X$$ ∂ X is homeomorphic to a manifold away from a set of codimension 2, and is $$N-1$$ N - 1 rectifiable. Along the way, we show effective measure bounds on the boundary and its tubular neighborhoods. These results are new even for Gromov–Hausdorff limits $$(M_i^N,{\mathsf {d}}_{g_i},p_i) \rightarrow (X,{\mathsf {d}},p)$$ ( M i N , d g i , p i ) → ( X , d , p ) of smooth manifolds with boundary, and require new techniques beyond those needed to prove the analogous statements for the regular set, in particular when it comes to the manifold structure of the boundary $$\partial X$$ ∂ X . The key local result is an $$\varepsilon $$ ε -regularity theorem, which tells us that if a ball $$B_{2}(p)\subset X$$ B 2 ( p ) ⊂ X is sufficiently close to a half space $$B_{2}(0)\subset {\mathbb {R}}^N_+$$ B 2 ( 0 ) ⊂ R + N in the Gromov–Hausdorff sense, then $$B_1(p)$$ B 1 ( p ) is biHölder to an open set of $${\mathbb {R}}^N_+$$ R + N . In particular, $$\partial X$$ ∂ X is itself homeomorphic to $$B_1(0^{N-1})$$ B 1 ( 0 N - 1 ) near $$B_1(p)$$ B 1 ( p ) . Further, the boundary $$\partial X$$ ∂ X is $$N-1$$ N - 1 rectifiable and the boundary measure "Equation missing" is Ahlfors regular on $$B_1(p)$$ B 1 ( p ) with volume close to the Euclidean volume. Our second collection of results involve the stability of the boundary with respect to noncollapsed mGH convergence $$X_i\rightarrow X$$ X i → X . Specifically, we show a boundary volume convergence which tells us that the $$N-1$$ N - 1 Hausdorff measures on the boundaries converge "Equation missing" to the limit Hausdorff measure on $$\partial X$$ ∂ X . We will see that a consequence of this is that if the $$X_i$$ X i are boundary free then so is X.

The metric measure boundary of spaces with Ricci curvature bounded below

“…(1.3) When S N −1 \ S N −2 = ∅, we say that X has no boundary. We refer to [BNS22] (see also the previous [DPG18,KM19]) for an account on regularity and stability of boundaries of RCD spaces.…”

“…The ingredient (3) comes from the recent [BNS22], see Theorem 2.2 for the precise statement and [JN16, CJN21, KLP21, LN20] for earlier versions in different contexts. It is used to globalize local bounds obtained out of (2) by summing up good scale invariant bounds on almost Euclidean balls.…”

Sharp isoperimetric comparison on non collapsed spaces with lower Ricci bounds

“…The uniform volume bound for the tubular neighbourhoods of ∂E follows from [20, Theorem 3.24], where quasiminimality of isoperimetric sets is proved, and [92, Lemma 2.42]. The conclusion follows arguing as in Step 1 of the proof of [92,Proposition 5.4] (see also the previous [32]).…”

“…Indeed, by the very definition, reduced boundary points for E are regular points of the ambient space (X, d, H N ). We refer to [53,70,32] for the relevant background about boundaries of RCD(K, N ) spaces (X, d, H N ) and just point out here that the notion is fully consistent with the case of smooth Riemannian manifolds and with the theory of Alexandrov spaces with sectional curvature bounded from below.…”

“…Compact convex subsets of Riemannian manifolds with Ricci curvature lower bounds have been considered in [27]. • The stability of mean curvature barriers in the sense of Theorem 1.2 for Gromov-Hausdorff converging sequences of boundaries inside measured Gromov-Hausdorff converging sequences of RCD(K, N ) spaces has been recently observed in [75] (see also the previous [32]). The main novelty of our work in this regard is to provide a large and natural class of sets having mean curvature barriers, namely isoperimetric sets.…”