Isometric immersions of RCD$(K,N)$ spaces via heat kernels
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866912170704371712 |
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| author | Huang, Zhangkai |
| author_facet | Huang, Zhangkai |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_11768 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Isometric immersions of RCD$(K,N)$ spaces via heat kernels Huang, Zhangkai Differential Geometry Metric Geometry 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. |
| title | Isometric immersions of RCD$(K,N)$ spaces via heat kernels |
| topic | Differential Geometry Metric Geometry |
| url | https://arxiv.org/abs/2205.11768 |