Boundary regularity and stability for spaces with Ricci bounded below

• 成果类型: Article
• 署名作者: Brue, Elia; Naber, Aaron; Semola, Daniele
• 署名单位: Institute for Advanced Study - USA; Northwestern University; University of Oxford
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-021-01092-8
• 发表日期: 2022
• 页码: 777-891
• 关键词: metric-measure-spaces; local poincare; tangent-cones; heat kernel; CURVATURE; MANIFOLDS; RIGIDITY; alexandrov; CONVERGENCE; stationary

摘要

This paper studies the structure and stability of boundaries in noncollapsed RCD(K,N) spaces, that is, metric-measure spaces (X,d,H-N) with Ricci curvature bounded below. Our main structural result is that the boundary partial derivative 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(gi), 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 derivative X. The key local result is an e-regularity theorem, which tells us that if a ball B-2(p) subset of X is sufficiently close to a half space B-2(0) subset of R-+(N) in the Gromov-Hausdorff sense, then B-1(p) is bi Holder to an open set of R-+(N). In particular, partial derivative X is itself homeomorphic to B-1(0(N-1)) near B-1(p). Further, the boundary partial derivative X is N-1 rectifiable and the boundary measure HN-1 (sic) partial derivative 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 -> X. Specifically, we show a boundary volume convergence which tells us that the N - 1 Hausdorff measures on the boundaries converge HN-1 (sic) partial derivative X-i. HN-1 (sic) partial derivative X to the limit Hausdorff measure on partial derivative X. We will see that a consequence of this is that if the Xi are boundary free then so is X.