Calculus of Variations and Geometric Measure Theory

E. Bruè - A. Naber - D. Semola

Boundary regularity and stability for spaces with Ricci bounded below

created by bruè on 17 Nov 2020
modified on 11 Feb 2022


Published Paper

Inserted: 17 nov 2020
Last Updated: 11 feb 2022

Journal: Inventiones mathematicae
Year: 2020
Doi: 10.1007/s00222-021-01092-8


This paper studies the structure and stability of boundaries in noncollapsed $RCD(K,N)$ spaces, that is, metric-measure spaces $(X,d,\mathscr{H}^N)$ with lower Ricci curvature bounded below. Our main structural result is that the boundary $\partial X$ is homeomorphic to a manifold away from a set of codimension 2, and is $N-1$ 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 $(M_i^N,d_{g_i},p_i) \rightarrow (X, d ,p)$ 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 $\partial X$.

The key local result is an $\epsilon$-regularity theorem, which tells us that if a ball $B_{2}(p)\subset X$ is sufficiently close to a half space $B_{2}(0)\subset \mathbb{R}^N_+$ in the Gromov-Hausdorff sense, then $B_1(p)$ is biH\"older to an open set of $\mathbb{R}^N_+$. In particular, $\partial X$ is itself homeomorphic to $B_1(0^{N-1})$ near $B_1(p)$. Further, the boundary $\partial X$ is $N-1$ rectifiable and the boundary measure $\mathscr{H}^{N-1}_{ \partial X}$ is Ahlfors regular on $B_1(p)$ with volume close to the Euclidean volume. Our second collection of results involve the stability of the boundary with respect to noncollapsed mGH convergence $X_i\to X$. Specifically, we show a boundary volume convergence which tells us that the $N-1$ Hausdorff measures on the boundaries converge $\mathscr{H}^{N-1}_{\partial X_i}\to \mathscr{H}^{N-1}_{\partial X}$ to the limit Hausdorff measure on $\partial X$. We will see that a consequence of this is that if the $X_i$ are boundary free then so is $X$.