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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.12292 |
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| _version_ | 1866913271558176768 |
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| author | Huang, G. Wang, X. -J. Zhou, Y. |
| author_facet | Huang, G. Wang, X. -J. Zhou, Y. |
| contents | In this paper, we prove the long time regularity of the interface in the $p$-Gauss curvature flow with flat side in all dimensions for $p>\frac1n$. Here the interface is the boundary of the flat part in the flow. In dimension $2$, this problem was solved in \cite{DL2004} for $p=1$ and in \cite{KimLeeRhee2013} for $p\in(1/2,1)$. We utilize the duality method to transform the Gauss curvature flow to a singular parabolic Monge-Ampère equation, and prove the regularity of the interface by studying the asymptotic cone of the parabolic Monge-Ampère equation in the polar coordinates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_12292 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Long time regularity of the $p$-Gauss curvature flow with flat side Huang, G. Wang, X. -J. Zhou, Y. Analysis of PDEs Differential Geometry In this paper, we prove the long time regularity of the interface in the $p$-Gauss curvature flow with flat side in all dimensions for $p>\frac1n$. Here the interface is the boundary of the flat part in the flow. In dimension $2$, this problem was solved in \cite{DL2004} for $p=1$ and in \cite{KimLeeRhee2013} for $p\in(1/2,1)$. We utilize the duality method to transform the Gauss curvature flow to a singular parabolic Monge-Ampère equation, and prove the regularity of the interface by studying the asymptotic cone of the parabolic Monge-Ampère equation in the polar coordinates. |
| title | Long time regularity of the $p$-Gauss curvature flow with flat side |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2403.12292 |