A Loewner-Nirenberg phenomena for Ricci flow on compact manifolds with boundary
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866915948815974400 |
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| author | Li, Gang |
| author_facet | Li, Gang |
| contents | In this paper, we show that starting from a geodesic ball $\overline{B_{r_0}}(0)$ in $\mathbb{H}^n$, for $n\geq3$, with prescribed non-decreasing rotationally symmetric mean curvature and the fixed conformal class $[g_{\mathbb{S}^{n-1}}]$ on the boundary, the solution $g(t)$ to the normalized Ricci flow $(1.2)$ which is continuous up to the boundary, exists for all $t>0$ and converges locally uniformly in $B_{r_0}(0)$ to a complete hyperbolic metric as $t\to\infty$(see Theorem 1.2 for details). Moreover, the sectional curvature of $g(t)$ maintains less than $-1$ for $t>0$. For dimension $2$, to achieve such a convergence result, we need the additional assumption that the mean curvature on the boundary increases in a certain speed to infinity as $t\to\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_20375 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Loewner-Nirenberg phenomena for Ricci flow on compact manifolds with boundary Li, Gang Differential Geometry Analysis of PDEs In this paper, we show that starting from a geodesic ball $\overline{B_{r_0}}(0)$ in $\mathbb{H}^n$, for $n\geq3$, with prescribed non-decreasing rotationally symmetric mean curvature and the fixed conformal class $[g_{\mathbb{S}^{n-1}}]$ on the boundary, the solution $g(t)$ to the normalized Ricci flow $(1.2)$ which is continuous up to the boundary, exists for all $t>0$ and converges locally uniformly in $B_{r_0}(0)$ to a complete hyperbolic metric as $t\to\infty$(see Theorem 1.2 for details). Moreover, the sectional curvature of $g(t)$ maintains less than $-1$ for $t>0$. For dimension $2$, to achieve such a convergence result, we need the additional assumption that the mean curvature on the boundary increases in a certain speed to infinity as $t\to\infty$. |
| title | A Loewner-Nirenberg phenomena for Ricci flow on compact manifolds with boundary |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2604.20375 |