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arXiv · 1101.3282

Constant mean curvature and totally umbilical biharmonic surfaces in 3-dimensional geometries

Ye-Lin Ou·Ze-Ping Wang
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Abstract

We prove that a totally umbilical biharmonic surface in any $3$-dimensional Riemannian manifold has constant mean curvature. We use this to show that a totally umbilical surface in Thurston's 3-dimensional geometries is proper biharmonic if and only if it is a part of $S^2(1/\sqrt{2})$ in $S^3$. We also give complete classifications of constant mean curvature proper biharmonic surfaces in 3-dimensional geometries and in 3-dimensional Bianchi-Cartan-Vranceanu spaces, and a complete classifications of proper biharmonic Hopf cylinders in 3-dimensional Bianchi-Cartan-Vranceanu spaces.

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Ye-Lin Ou, Ze-Ping Wang. 2011-01-17. Constant mean curvature and totally umbilical biharmonic surfaces in 3-dimensional geometries. https://doi.org/10.1016/j.geomphys.2011.04.008

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