Isotropy invariant graphical mean curvature flows in warped products
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916523487002624 |
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| author | Fujihara, Naotoshi Koike, Naoyuki |
| author_facet | Fujihara, Naotoshi Koike, Naoyuki |
| contents | In this paper, we study the graphical mean curvature flow in a warped product $_r G/K \times I$, where $G/K$ is a symmetric space of compact type, $I$ is an open interval, and $r$ is a smooth positive function on $I$. If the initial hypersurface is $K$-equivariant, then the $K$-equivariance is preserved along the mean curvature flow. Here, we note that isotropy group $K$ acts naturally on both $G/K$ and $_r G/K \times I$. If the flow is graphical, then it follows from the $K$-equivariance of the flow that it can be described by using $K$-invariant functions on $G/K$. We derive the flow equation which these functions satisfy. By using the flow equation, we prove that the mean curvature flow exists for infinite time under the conditions that $G/K$ is a rank one symmetric space of compact type and the warping function $r$ satisfies certain additional properties. The proof is carried out by estimating the gradient of the $K$-invariant functions satisfying the flow equation. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2412_10711 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Isotropy invariant graphical mean curvature flows in warped products Fujihara, Naotoshi Koike, Naoyuki Differential Geometry 53E10 (Primary), 53C40, 53C35 (Secondary) In this paper, we study the graphical mean curvature flow in a warped product $_r G/K \times I$, where $G/K$ is a symmetric space of compact type, $I$ is an open interval, and $r$ is a smooth positive function on $I$. If the initial hypersurface is $K$-equivariant, then the $K$-equivariance is preserved along the mean curvature flow. Here, we note that isotropy group $K$ acts naturally on both $G/K$ and $_r G/K \times I$. If the flow is graphical, then it follows from the $K$-equivariance of the flow that it can be described by using $K$-invariant functions on $G/K$. We derive the flow equation which these functions satisfy. By using the flow equation, we prove that the mean curvature flow exists for infinite time under the conditions that $G/K$ is a rank one symmetric space of compact type and the warping function $r$ satisfies certain additional properties. The proof is carried out by estimating the gradient of the $K$-invariant functions satisfying the flow equation. |
| title | Isotropy invariant graphical mean curvature flows in warped products |
| topic | Differential Geometry 53E10 (Primary), 53C40, 53C35 (Secondary) |
| url | https://arxiv.org/abs/2412.10711 |