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

We prove that a totally umbilical biharmonic surface in any 33-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 S2(1/2) in S3S3. We also give complete classifications of constant mean curvature proper biharmonic surfaces in Thurston’s 33-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.