Willmore surfaces and Hopf tori in homogeneous 3-manifolds
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910321122213888 |
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| author | Albujer, Alma L. Santos, Fábio R. dos |
| author_facet | Albujer, Alma L. Santos, Fábio R. dos |
| contents | Some classification results for closed surfaces in Berger spheres are presented. On the one hand, a Willmore functional for isometrically immersed surfaces into an homogeneous space $\mathbb{E}^{3}(κ,τ)$ with isometry group of dimension $4$ is defined and its first variational formula is computed. Then, we characterize Clifford and Hopf tori as the only Willmore surfaces satifying a sharp Simons-type integral inequality. On the other hand, we also obtain some integral inequalities for closed surfaces with constant extrinsic curvature in $\mathbb{E}^3(κ,τ)$, becoming equalities if and only if the surface is a Hopf torus in a Berger sphere. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2402_04654 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Willmore surfaces and Hopf tori in homogeneous 3-manifolds Albujer, Alma L. Santos, Fábio R. dos Differential Geometry 53C42, 53A10, 53C30 Some classification results for closed surfaces in Berger spheres are presented. On the one hand, a Willmore functional for isometrically immersed surfaces into an homogeneous space $\mathbb{E}^{3}(κ,τ)$ with isometry group of dimension $4$ is defined and its first variational formula is computed. Then, we characterize Clifford and Hopf tori as the only Willmore surfaces satifying a sharp Simons-type integral inequality. On the other hand, we also obtain some integral inequalities for closed surfaces with constant extrinsic curvature in $\mathbb{E}^3(κ,τ)$, becoming equalities if and only if the surface is a Hopf torus in a Berger sphere. |
| title | Willmore surfaces and Hopf tori in homogeneous 3-manifolds |
| topic | Differential Geometry 53C42, 53A10, 53C30 |
| url | https://arxiv.org/abs/2402.04654 |