Boundary regularity and stability for spaces with Ricci bounded below

Invent. math. https://doi.org/10.1007/s00222-021-01092-8 Boundary regularity and stability for spaces with Ricci bounded below Elia Bruè1 · Aaron Naber2 · Daniele Semola3 Received: 30 November 2020 / Accepted: 30 November 2021 © Crown 2022 Abstract 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 ∂ 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 (MiN , dgi , pi ) → (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 ∂ X . The key local result is an ε-regularity theorem, which tells us that if a ball B2 ( p) ⊂ X is sufficiently close to a half N in the Gromov–Hausdorff sense, then B ( p) is biHölder to space B2 (0) ⊂ R+ 1 N an open set of R+ . In particular, ∂ X is itself homeomorphic to B1 (0 N −1 ) near B1 ( p). Further, the boundary ∂ X is N −1 rectifiable and the boundary measure H N −1 ∂ X is Ahlfors regular on B1 ( p) with volume close to the Euclidean B Daniele Semola Elia Bruè Aaron Naber 1 School of Mathematics, Institute for Advanced Study, 1 Einstein Dr., Princeton, NJ 05840, USA 2 Northwestern University, 633 Clark Street, Evanston, IL 60208, USA 3 Mathematical Institute, University of Oxford, Oxford OX2 6GG, UK 123 E. Bruè et al. 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 H N −1 ∂ X i → H N −1 ∂ X to the limit Hausdorff measure on ∂ X . We will see that a consequence of this is that if the X i are boundary free then so is X . Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Singular strata and boundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 An ε-regularity theorem for top dimensional singularities . . . . . . . . . . . . . . . 1.3 Structure of boundaries and of spaces with boundary . . . . . . . . . . . . . . . . . 1.4 Stability and gap theorems for boundaries . . . . . . . . . . . . . . . . . . . . . . . 1.5 Comparison with the Alexandrov theory . . . . . . . . . . . . . . . . . . . . . . . . 1.6 The remainder of the paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Calculus tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 RCD spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Structure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Calculus on RCD spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Continuity equation and flow maps . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Noncollapsed spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Cone splitting via content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Splitting maps on RCD spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Functional splitting theorem, local version . . . . . . . . . . . . . . . . . . . . . . . 3.2 δ-Splitting maps and ε-GH isometries . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Transformation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Neck regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Structure of neck regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Existence of neck regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Auxiliary results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Strategy of proof of Theorem 4.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Proof of Proposition 4.19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Proof of Theorem 4.13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Neck Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Proof of the neck decomposition theorem . . . . . . . . . . . . . . . . . . . . . . . 6 Boundary rectifiability and stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Proof of the stability results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Rectifiable structure and volume estimates . . . . . . . . . . . . . . . . . . . . . . . 6.3 A second notion of boundary and further regularity properties . . . . . . . . . . . . . 7 Distance from the boundary and noncollapsing of boundaries . . . . . . . . . . . . . . . 7.1 Laplacian of the distance from the boundary . . . . . . . . . . . . . . . . . . . . . . 7.2 Alexandrov spaces and noncollapsed Ricci limits with boundary . . . . . . . . . . . 8 Improved neck structure and boundary measure convergence . . . . . . . . . . . . . . . . 8.1 Improved neck structure theorem and boundary volume rigidity . . . . . . . . . . . . 8.2 Topological regularity of the boundary . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 Convergence of boundary measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Topological regularity up to the boundary . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 Boundary regularity and stability for spaces 1 Introduction This paper studies structural and stability properties for noncollapsed RCD(K , N ) spaces with boundary. In particular, we give affirmative answers to some of the recent conjectures presented in [37,59]. Most of the statements are new and of interest even for noncollapsed limits of smooth Riemannian manifolds with convex boundary and interior lower Ricci curvature bounds. Our main results can be grouped into • Structure results for boundaries and spaces with boundary; • Stability/gap theorems about the absence/presence of boundary. In particular, we obtain the rectifiable structure of the boundary together with measure estimates. Moreover we prove that noncollapsed RCD spaces are homeomorphic to topological manifolds (possibly with boundary) up to sets of codimension two. On the side of stability/gap results we are going to prove that the absence of boundary is preserved under noncollapsed (pointed) Gromov–Hausdorff convergence and that the boundary volume measures converg (...truncated)