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arXiv · 2410.20517

$f$-Biharmonic hypersurfaces into a conformally flat space

Ze-Ping Wang·Li-Hua Qin·Xue-Yi Chen
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Abstract

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.

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Ze-Ping Wang, Li-Hua Qin, Xue-Yi Chen. 2024-10-27. $f$-Biharmonic hypersurfaces into a conformally flat space. https://arxiv.org/abs/2410.20517

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