A Geometric flow towards hamiltonian stationary submanifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866912041701212160 |
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| author | Chen, Jingyi Warren, Micah |
| author_facet | Chen, Jingyi Warren, Micah |
| contents | In this paper, we introduce a geometric flow for Lagrangian submanifolds in a Kähler manifold that stays in its initial Hamiltonian isotopy class and is a gradient flow for volume. The stationary solutions are the Hamiltonian stationary Lagrangian submanifolds. The flow is not strictly parabolic but it corresponds to a fourth order strictly parabolic scalar equation in the cotangent bundle of the submanifold via Weinstein's Lagrangian neighborhood theorem. For any compact initial Lagrangian immersion, we establish short-time existence, uniqueness, and higher order estimates when the second fundamental forms are uniformly bounded up to time $T$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_13997 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Geometric flow towards hamiltonian stationary submanifolds Chen, Jingyi Warren, Micah Differential Geometry Analysis of PDEs 53E40, 58J35, 35K25 In this paper, we introduce a geometric flow for Lagrangian submanifolds in a Kähler manifold that stays in its initial Hamiltonian isotopy class and is a gradient flow for volume. The stationary solutions are the Hamiltonian stationary Lagrangian submanifolds. The flow is not strictly parabolic but it corresponds to a fourth order strictly parabolic scalar equation in the cotangent bundle of the submanifold via Weinstein's Lagrangian neighborhood theorem. For any compact initial Lagrangian immersion, we establish short-time existence, uniqueness, and higher order estimates when the second fundamental forms are uniformly bounded up to time $T$. |
| title | A Geometric flow towards hamiltonian stationary submanifolds |
| topic | Differential Geometry Analysis of PDEs 53E40, 58J35, 35K25 |
| url | https://arxiv.org/abs/2403.13997 |