Second boundary value problem for the Hessian curvature flow
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866915311928737792 |
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| author | Huang, Rongli Qu, Changzheng Wang, Zhizhang Wo, Weifeng |
| author_facet | Huang, Rongli Qu, Changzheng Wang, Zhizhang Wo, Weifeng |
| contents | We investigate the evolution of strictly convex hypersurfaces driven by the $k$-Hessian curvature flow, subject to the second boundary condition. We first explore the translating solutions corresponding to this boundary value problem. Next, we establish the long-time existence of the flow and prove that it converges to a translating solution. To overcome the difficulty of driving boundary $C^2$ estimates, we employ an orthogonal invariance technique. Using this method, we extend the results of Schnürer-Smoczyk \cite{Schnurer2003} and Schnürer \cite{Schnurer2002} from the second boundary value problem of Gauss curvature flow to $k$-Hessian curvature flow. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_23067 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Second boundary value problem for the Hessian curvature flow Huang, Rongli Qu, Changzheng Wang, Zhizhang Wo, Weifeng Analysis of PDEs We investigate the evolution of strictly convex hypersurfaces driven by the $k$-Hessian curvature flow, subject to the second boundary condition. We first explore the translating solutions corresponding to this boundary value problem. Next, we establish the long-time existence of the flow and prove that it converges to a translating solution. To overcome the difficulty of driving boundary $C^2$ estimates, we employ an orthogonal invariance technique. Using this method, we extend the results of Schnürer-Smoczyk \cite{Schnurer2003} and Schnürer \cite{Schnurer2002} from the second boundary value problem of Gauss curvature flow to $k$-Hessian curvature flow. |
| title | Second boundary value problem for the Hessian curvature flow |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2505.23067 |