论文标题
RCD $（k，n）$空间的等距沉浸式通过热内核
Isometric immersions of RCD$(K,N)$ spaces via heat kernels
论文作者
论文摘要
给定RCD $（k，n）$ space $（{x}，\ mathsf {d}，\ mathfrak {m}）$，可以使用其热核$ρ$将其映射到$ l^2 $ space中，该$ l^2 $ space由本地lipslylylipslylylipslylipslylipslylipshitz map $φ_t（x）：x =ρ（x）：= x（x，x，cdot，cdot，t）。空间$（x，x，\ mathsf {d}，\ mathfrak {m}）$被认为是同等的加热内核浸入空间，如果每个$φ_T$都是等距沉浸式{} {正常化后}。一个主要结果指出，任何紧凑的等二含量加热内核浸入RCD $（k，n）$空间均与未加权的闭合光滑的riemannian歧管相等。这是一个更普遍的结果证明的：如果紧凑的非cd $（k，n）$ space具有等速度沉浸的特征玛普，那么该空间与未加权的封闭的利马尼亚歧管等均等，这在\ cite {h21}中的规律性大大改善了{H21}的规律性。作为这些结果的应用，我们为特定类别具有曲率二次绑定的Riemannian歧管给出了$ c^\ infty $ compactness定理，并且具有等值的特征性特征。
Given an RCD$(K,N)$ space $({X},\mathsf{d},\mathfrak{m})$, one can use its heat kernel $ρ$ to map it into the $L^2$ space by a locally Lipschitz map $Φ_t(x):=ρ(x,\cdot,t)$. The space $(X,\mathsf{d},\mathfrak{m})$ is said to be an isometrically heat kernel immersing space, if each $Φ_t$ is an isometric immersion {}{after a normalization}. A main result states that any compact isometrically heat kernel immersing RCD$(K,N)$ space is isometric to an unweighted closed smooth Riemannian manifold. This is justified by a more general result: if a compact non-collapsed RCD$(K, N)$ space has an isometrically immersing eigenmap, then the space is isometric to an unweighted closed Riemannian manifold, which greatly improves a regularity result in \cite{H21} by Honda. As an application of these results, we give a $C^\infty$-compactness theorem for a certain class of Riemannian manifolds with a curvature-dimension-diameter bound and an isometrically immersing eigenmap.
