New exotic examples of Ricci limit spaces
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866913546358489088 |
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| author | Li, Xilun Zhou, Shengxuan |
| author_facet | Li, Xilun Zhou, Shengxuan |
| contents | For any integers $m\geqslant n\geqslant 3$, we construct a Ricci limit space $X_{m,n}$ such that for a fixed point, some tangent cones are $\mathbb{R}^m$ and some are $\mathbb{R}^n$. This is an improvement of Menguy's example. Moreover, we show that for any finite collection of closed Riemannian manifolds $(M_i^{n_i},g_i)$ with $\mathrm{Ric}_{g_i}\geqslant(n_i-1)\geqslant 1$, there exists a collapsed Ricci limit space $(X,d,x)$ such that each Riemannian cone $C(M_i,g_i)$ is a tangent cone of $X$ at $x$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_15054 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | New exotic examples of Ricci limit spaces Li, Xilun Zhou, Shengxuan Differential Geometry Metric Geometry For any integers $m\geqslant n\geqslant 3$, we construct a Ricci limit space $X_{m,n}$ such that for a fixed point, some tangent cones are $\mathbb{R}^m$ and some are $\mathbb{R}^n$. This is an improvement of Menguy's example. Moreover, we show that for any finite collection of closed Riemannian manifolds $(M_i^{n_i},g_i)$ with $\mathrm{Ric}_{g_i}\geqslant(n_i-1)\geqslant 1$, there exists a collapsed Ricci limit space $(X,d,x)$ such that each Riemannian cone $C(M_i,g_i)$ is a tangent cone of $X$ at $x$. |
| title | New exotic examples of Ricci limit spaces |
| topic | Differential Geometry Metric Geometry |
| url | https://arxiv.org/abs/2404.15054 |