Entropy and stability of hyperbolic manifolds
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arXiv
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866916922060177408 |
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| author | Song, Antoine |
| author_facet | Song, Antoine |
| contents | Let $(M,g_0)$ be a closed oriented hyperbolic manifold of dimension at least $3$. By the volume entropy inequality of G. Besson, G. Courtois and S. Gallot, for any Riemannian metric $g$ on $M$ with same volume as $g_0$, its volume entropy $h(g)$ satisfies $h(g)\geq n-1$ with equality only when $g$ is isometric to $g_0$. We show that the hyperbolic metric $g_0$ is stable in the following sense: if $g_i$ is a sequence of Riemaniann metrics on $M$ of same volume as $g_0$ and if $h(g_i)$ converges to $n-1$, then there are smooth subsets $Z_i\subset M$ such that both $\mathrm{Vol}(Z_i,g_i)$ and $\mathrm{Area}(\partial Z_i,g_i)$ tend to $0$, and $(M\setminus Z_i,g_i)$ converges to $(M,g_0)$ in the measured Gromov-Hausdorff topology. The proof relies on showing that any spherical Plateau solution for $M$ is intrinsically isomorphic to $(M,\frac{(n-1)^2}{4n} g_0)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_07422 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Entropy and stability of hyperbolic manifolds Song, Antoine Differential Geometry Let $(M,g_0)$ be a closed oriented hyperbolic manifold of dimension at least $3$. By the volume entropy inequality of G. Besson, G. Courtois and S. Gallot, for any Riemannian metric $g$ on $M$ with same volume as $g_0$, its volume entropy $h(g)$ satisfies $h(g)\geq n-1$ with equality only when $g$ is isometric to $g_0$. We show that the hyperbolic metric $g_0$ is stable in the following sense: if $g_i$ is a sequence of Riemaniann metrics on $M$ of same volume as $g_0$ and if $h(g_i)$ converges to $n-1$, then there are smooth subsets $Z_i\subset M$ such that both $\mathrm{Vol}(Z_i,g_i)$ and $\mathrm{Area}(\partial Z_i,g_i)$ tend to $0$, and $(M\setminus Z_i,g_i)$ converges to $(M,g_0)$ in the measured Gromov-Hausdorff topology. The proof relies on showing that any spherical Plateau solution for $M$ is intrinsically isomorphic to $(M,\frac{(n-1)^2}{4n} g_0)$. |
| title | Entropy and stability of hyperbolic manifolds |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2302.07422 |