Boundary regularity and stability for spaces with Ricci bounded below
Boundary regularity and stability for spaces with Ricci bounded below
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DOI：
10.1007/s00222-021-01092-8
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发表时间：
2020-11
影响因子：
3.1
通讯作者：
Elia Brué;A. Naber;Daniele Semola
中科院分区：
文献类型：
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作者：
Elia Brué;A. Naber;Daniele Semola
本文研究了非塌陷RCD(K,N)空间中边界的结构和稳定性，即Ricci曲率下界的度量-测度空间(X,d,HN)。我们的主要结构结果是边界 ∂ X 同胚于远离余维 2 集合的流形，并且是 N-1 可整流的。在此过程中，我们展示了边界及其管状邻域的有效测度边界。即使对于具有边界的光滑流形的 Gromov–Hausdorff 极限 (M i N, dgi, pi)→(X, d, p) 来说，这些结果也是新的，并且需要超出证明正则集的类似陈述所需的新技术，特别是当涉及边界 ∂ X 的流形结构时。关键的局部结果是 ε-正则定理，它告诉我们如果球 B 2 (p)⊂ X 足够接近半空间在格罗莫夫-豪斯多夫意义上，B 2 (0)⊂ R+ N，则 B 1 (p) 是 R+ N 开集的 biHölder。特别是，∂ X 本身同胚于 B 1 (p) 附近的 B 1 (0 N-1)。此外，边界∂ X 是 N-1 可校正的，并且边界测度是 B 1 (p) 上的 Ahlfors 正则，体积接近欧几里得体积。我们的第二组结果涉及关于非塌缩 mGH 收敛 X i→ X 的边界稳定性。具体来说，我们显示了边界体积收敛，它告诉我们边界上的 N-1 Hausdorff 测度收敛到 ∂ X 上的极限 Hausdorff 测度。我们将看到，这样做的结果是，如果 X i 是边界自由的，那么 X 也是边界自由的。
This paper studies the structure and stability of boundaries in noncollapsed RCD (K, N) spaces, that is, metric-measure spaces (X, d, HN) 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 (M i N, 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 B 2 (p)⊂ X is sufficiently close to a half space B 2 (0)⊂ R+ N in the Gromov–Hausdorff sense, then B 1 (p) is biHölder to an open set of R+ N. In particular,∂ X is itself homeomorphic to B 1 (0 N-1) near B 1 (p). Further, the boundary∂ X is N-1 rectifiable and the boundary measure 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 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.