$f$-Biharmonic hypersurfaces into a conformally flat space
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916456758771712 |
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| author | Wang, Ze-Ping Qin, Li-Hua Chen, Xue-Yi |
| author_facet | Wang, Ze-Ping Qin, Li-Hua Chen, Xue-Yi |
| contents | We first study $f$-biharmonicity of totally umbilical hypersurfaces in a generic Riemannian manifold and then prove that any totally umbilical proper $f$-biharmonic hypersurface in a nonpositively curved manifold has to be noncompact. We also explore $f$-biharmonicity of totally umbilical hyperplanes in a conformally flat space. Secondly, we construct $f$-biharmonic surfaces and biharmonic conformal immersions of the associated surfaces into a conformall flat 3-space and also give a complete classification of $f$-biharmonic surfaces of nonzero constant mean curvature in 3-space forms. Finally, we especially investigate $f$-biharmonicity of hypersurfaces into a conformally flat space of negative sectional curvature. We show that any totally umbilical $f$-biharmonic surface of a 3-manifold with nonpositve sectional curvature is minimal whilst there are proper $f$-biharmonic $m$-dimensional submanifolds with $m\geq3$ and $m\neq4$ into nonpositvely curved manifolds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_20517 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $f$-Biharmonic hypersurfaces into a conformally flat space Wang, Ze-Ping Qin, Li-Hua Chen, Xue-Yi Differential Geometry We first study $f$-biharmonicity of totally umbilical hypersurfaces in a generic Riemannian manifold and then prove that any totally umbilical proper $f$-biharmonic hypersurface in a nonpositively curved manifold has to be noncompact. We also explore $f$-biharmonicity of totally umbilical hyperplanes in a conformally flat space. Secondly, we construct $f$-biharmonic surfaces and biharmonic conformal immersions of the associated surfaces into a conformall flat 3-space and also give a complete classification of $f$-biharmonic surfaces of nonzero constant mean curvature in 3-space forms. Finally, we especially investigate $f$-biharmonicity of hypersurfaces into a conformally flat space of negative sectional curvature. We show that any totally umbilical $f$-biharmonic surface of a 3-manifold with nonpositve sectional curvature is minimal whilst there are proper $f$-biharmonic $m$-dimensional submanifolds with $m\geq3$ and $m\neq4$ into nonpositvely curved manifolds. |
| title | $f$-Biharmonic hypersurfaces into a conformally flat space |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2410.20517 |