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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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