Rigidity and regularity for almost homogeneous spaces with Ricci curvature bounds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913628849963008 |
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| author | Qian, Xin |
| author_facet | Qian, Xin |
| contents | We say that a metric space $X$ is $(ε,G)$-homogeneous if $G<Iso(X)$ is a discrete group of isometries with $diam(X/G)<ε$.\ A sequence of $(ε_i,G_i)$-homogeneous spaces $X_i$ with $ε_i\to0$ is called a sequence of almost homogeneous spaces.
In this paper we show that the Gromov-Hausdorff limit of a sequence of almost homogeneous RCD$(K,N)$ spaces must be a nilpotent Lie group with $Ric\geqslant K$. We also obtain a topological rigidity theorem for $(ε,G)$-homogeneous RCD$(K,N)$ spaces, which generalizes a recent result by Wang. Indeed, if $X$ is an $(ε,G)$-homogeneous RCD$(K,N)$ space and $G$ is an almost-crystallographic group, then $X/G$ is bi-Hölder to an infranil orbifold. Moreover, we study $(ε,G)$-homogeneous spaces in the smooth setting and prove rigidity and $ε$-regularity theorems for Riemannian orbifolds with Einstein metrics and bounded Ricci curvatures respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20353 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Rigidity and regularity for almost homogeneous spaces with Ricci curvature bounds Qian, Xin Differential Geometry Metric Geometry We say that a metric space $X$ is $(ε,G)$-homogeneous if $G<Iso(X)$ is a discrete group of isometries with $diam(X/G)<ε$.\ A sequence of $(ε_i,G_i)$-homogeneous spaces $X_i$ with $ε_i\to0$ is called a sequence of almost homogeneous spaces. In this paper we show that the Gromov-Hausdorff limit of a sequence of almost homogeneous RCD$(K,N)$ spaces must be a nilpotent Lie group with $Ric\geqslant K$. We also obtain a topological rigidity theorem for $(ε,G)$-homogeneous RCD$(K,N)$ spaces, which generalizes a recent result by Wang. Indeed, if $X$ is an $(ε,G)$-homogeneous RCD$(K,N)$ space and $G$ is an almost-crystallographic group, then $X/G$ is bi-Hölder to an infranil orbifold. Moreover, we study $(ε,G)$-homogeneous spaces in the smooth setting and prove rigidity and $ε$-regularity theorems for Riemannian orbifolds with Einstein metrics and bounded Ricci curvatures respectively. |
| title | Rigidity and regularity for almost homogeneous spaces with Ricci curvature bounds |
| topic | Differential Geometry Metric Geometry |
| url | https://arxiv.org/abs/2412.20353 |