论文标题
从RCD空间到Riemannian歧管的拓扑稳定性定理的注释
A note on the topological stability theorem from RCD spaces to Riemannian manifolds
论文作者
论文摘要
Inspired by a recent work of Wang-Zhao, in this note we prove that for a fixed $n$-dimensional closed Riemannian manifold $(M^n, g)$, if an $\mathrm{RCD}(K, n)$ space $(X, \mathsf{d}, \mathfrak{m})$ is Gromov-Hausdorff close to $M^n$, then存在常规同构$ f $从$ x $到$ m^n $，使得$ f $是Lipschitz的连续，并且$ f^{ - 1} $是hölder的连续，在这里，$ f $，$ f $，hölder指数的lipschitz常数，hölder指数和$ f^{ - 1} $可以$ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $ 1 $。从某种意义上说，这是一个敏锐的意义，即这样的地图不能改善为Bi-lipschitz。此外，如果$ x $平稳，那么可以选择这样的同态形态作为差异性。值得一提的是，$ f $的Lipschitz-Hölder连续性改善了固有的Reifenberg Theorem的封闭歧管定理，而Cheeger-Colding建立了RICCI曲率，而Cheeger-colding则建立了RICCI曲率。纳什嵌入定理在证明中起关键作用。
Inspired by a recent work of Wang-Zhao, in this note we prove that for a fixed $n$-dimensional closed Riemannian manifold $(M^n, g)$, if an $\mathrm{RCD}(K, n)$ space $(X, \mathsf{d}, \mathfrak{m})$ is Gromov-Hausdorff close to $M^n$, then there exists a regular homeomorphism $F$ from $X$ to $M^n$ such that $F$ is Lipschitz continuous and that $F^{-1}$ is Hölder continuous, where the Lipschitz constant of $F$, the Hölder exponent and the Hölder constant of $F^{-1}$ can be chosen arbitrary close to $1$. This is sharp in the sense that in general such a map cannot be improved to being bi-Lipschitz. Moreover if $X$ is smooth, then such a homeomorphism can be chosen as a diffeomorphism. It is worth mentioning that the Lipschitz-Hölder continuity of $F$ improves the intrinsic Reifenberg theorem for closed manifolds with Ricci curvature bounded below established by Cheeger-Colding. The Nash embedding theorem plays a key role in the proof.
