Eigenvalue problems and free boundary minimal surfaces in spherical caps
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866909328037904384 |
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| author | Lima, Vanderson Menezes, Ana |
| author_facet | Lima, Vanderson Menezes, Ana |
| contents | Given a compact surface with boundary, we introduce a family of functionals on the space of its Riemannian metrics, defined via eigenvalues of a Steklov-type problem. We prove that each such functional is uniformly bounded from above, and we characterize maximizing metrics as induced by free boundary minimal immersions in some geodesic ball of a round sphere. Also, we determine that the maximizer in the case of a disk is a spherical cap of dimension two, and we prove rotational symmetry of free boundary minimal annuli in geodesic balls of round spheres which are immersed by first eigenfunctions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_13556 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Eigenvalue problems and free boundary minimal surfaces in spherical caps Lima, Vanderson Menezes, Ana Differential Geometry Given a compact surface with boundary, we introduce a family of functionals on the space of its Riemannian metrics, defined via eigenvalues of a Steklov-type problem. We prove that each such functional is uniformly bounded from above, and we characterize maximizing metrics as induced by free boundary minimal immersions in some geodesic ball of a round sphere. Also, we determine that the maximizer in the case of a disk is a spherical cap of dimension two, and we prove rotational symmetry of free boundary minimal annuli in geodesic balls of round spheres which are immersed by first eigenfunctions. |
| title | Eigenvalue problems and free boundary minimal surfaces in spherical caps |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2307.13556 |