On weak inverse mean curvature flow and Minkowski-type inequalities in hyperbolic space
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866911970422161408 |
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| author | Harvie, Brian |
| author_facet | Harvie, Brian |
| contents | We prove that a proper weak solution $\{ Ω_{t} \}_{0 \leq t < \infty}$ to inverse mean curvature flow in $\mathbb{H}^{n}$, $3\leq n \leq 7$, is smooth and star-shaped by the time
\begin{equation*}
T= (n-1) \log \left( \frac{\text{sinh} \left( r_{+} \right)}{ \text{sinh} \left( r_{-} \right)} \right), \end{equation*} where $r_{+}$ and $r_{-}$ are the geodesic out-radius and in-radius of the initial domain $Ω_{0}$. The argument is inspired by the Alexandrov reflection method for extrinsic curvature flows in $\mathbb{R}^{n}$ due to Chow-Gulliver and uses a result of Li-Wei. In addition to this, our methods establish expanding spheres as the only proper weak IMCF on $\mathbb{H}^{n} \setminus \{ 0 \}$ in all dimensions.
As applications, we extend the Minkowski inequalities of Brendle-Hung-Wang and De Lima-Girao to outer-minimizing domains $Ω_{0} \subset \mathbb{H}^{n}$ in dimensions $3 \leq n \leq 7$. From this, we also extend a Penrose-type inequality to balanced asymptotically hyperbolic graphs over the exteriors of outer-minimizing domains of $\mathbb{H}^{n}$ in these dimensions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08410 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On weak inverse mean curvature flow and Minkowski-type inequalities in hyperbolic space Harvie, Brian Differential Geometry Analysis of PDEs We prove that a proper weak solution $\{ Ω_{t} \}_{0 \leq t < \infty}$ to inverse mean curvature flow in $\mathbb{H}^{n}$, $3\leq n \leq 7$, is smooth and star-shaped by the time \begin{equation*} T= (n-1) \log \left( \frac{\text{sinh} \left( r_{+} \right)}{ \text{sinh} \left( r_{-} \right)} \right), \end{equation*} where $r_{+}$ and $r_{-}$ are the geodesic out-radius and in-radius of the initial domain $Ω_{0}$. The argument is inspired by the Alexandrov reflection method for extrinsic curvature flows in $\mathbb{R}^{n}$ due to Chow-Gulliver and uses a result of Li-Wei. In addition to this, our methods establish expanding spheres as the only proper weak IMCF on $\mathbb{H}^{n} \setminus \{ 0 \}$ in all dimensions. As applications, we extend the Minkowski inequalities of Brendle-Hung-Wang and De Lima-Girao to outer-minimizing domains $Ω_{0} \subset \mathbb{H}^{n}$ in dimensions $3 \leq n \leq 7$. From this, we also extend a Penrose-type inequality to balanced asymptotically hyperbolic graphs over the exteriors of outer-minimizing domains of $\mathbb{H}^{n}$ in these dimensions. |
| title | On weak inverse mean curvature flow and Minkowski-type inequalities in hyperbolic space |
| topic | Differential Geometry Analysis of PDEs |
| url | https://arxiv.org/abs/2404.08410 |