On the non-collapsed RCD spaces with local bounded covering geometry
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912149204369408 |
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| author | Wang, Jikang |
| author_facet | Wang, Jikang |
| contents | We consider a RCD$(-(N-1),N)$ space $(X,d,\mathcal{H}^N)$ with local bounded covering geometry. The first result is related to Gromov's almost flat manifold theorem. Specifically, if for every point $\tilde{p}$ in the universal cover $\widetilde{X}$, we have $\mathcal{H}^N(B_1(\tilde{p})) \ge v > 0$ and the diameter of $X$ is sufficiently small, then $X$ is biHölder homeomorphic to an infranil-manifold. Moreover, if $X$ is a smooth Riemannian $N$-manifold with $\mathrm{Ric} \ge -(N-1)$, then $X$ is biHölder diffeomorphic to an infranil-manifold. An application of our argument is to confirm the conjecture that Gromov's almost flat manifold theorem holds in the $\mathrm{RCD}+\mathrm{CBA}$ setting.
The second result concerns a regular fibration theorem. Let $(X_i,d_i,\mathcal{H}^N)$ be a sequence of RCD$(-(N-1),N)$ spaces converging to a compact smooth $k$-dimensional manifold $K$ in the Gromov-Hausdorff sense. Assume that for any $p_i \in X_i$, the local universal cover is non-collapsing, i.e., for any pre-image point $\tilde{p}_i$ of $p_i$ in the universal cover of the ball $B_3(p_i)$, we have $\mathcal{H}^N(B_{1}(\tilde{p}_i)) \ge v$ for some fixed $v>0$. Then for sufficiently large $i$, there exists a fibration map $f_i:X_i \to K$, where the fiber is an infra-nilmanifold and the structure group is affine. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_06131 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the non-collapsed RCD spaces with local bounded covering geometry Wang, Jikang Differential Geometry Metric Geometry We consider a RCD$(-(N-1),N)$ space $(X,d,\mathcal{H}^N)$ with local bounded covering geometry. The first result is related to Gromov's almost flat manifold theorem. Specifically, if for every point $\tilde{p}$ in the universal cover $\widetilde{X}$, we have $\mathcal{H}^N(B_1(\tilde{p})) \ge v > 0$ and the diameter of $X$ is sufficiently small, then $X$ is biHölder homeomorphic to an infranil-manifold. Moreover, if $X$ is a smooth Riemannian $N$-manifold with $\mathrm{Ric} \ge -(N-1)$, then $X$ is biHölder diffeomorphic to an infranil-manifold. An application of our argument is to confirm the conjecture that Gromov's almost flat manifold theorem holds in the $\mathrm{RCD}+\mathrm{CBA}$ setting. The second result concerns a regular fibration theorem. Let $(X_i,d_i,\mathcal{H}^N)$ be a sequence of RCD$(-(N-1),N)$ spaces converging to a compact smooth $k$-dimensional manifold $K$ in the Gromov-Hausdorff sense. Assume that for any $p_i \in X_i$, the local universal cover is non-collapsing, i.e., for any pre-image point $\tilde{p}_i$ of $p_i$ in the universal cover of the ball $B_3(p_i)$, we have $\mathcal{H}^N(B_{1}(\tilde{p}_i)) \ge v$ for some fixed $v>0$. Then for sufficiently large $i$, there exists a fibration map $f_i:X_i \to K$, where the fiber is an infra-nilmanifold and the structure group is affine. |
| title | On the non-collapsed RCD spaces with local bounded covering geometry |
| topic | Differential Geometry Metric Geometry |
| url | https://arxiv.org/abs/2412.06131 |