## Constant mean curvature and totally umbilical biharmonic surfaces in 3-dimensional geometries.(English)Zbl 1227.58004

Summary: 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 Thurston’s 3-dimensional geometries and in 3-dimensional Bianchi-Cartan-Vranceanu spaces, and a complete classification of proper biharmonic Hopf cylinders in 3-dimensional Bianchi-Cartan-Vranceanu spaces.

### MSC:

 58E20 Harmonic maps, etc. 53C12 Foliations (differential geometric aspects) 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
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### References:

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