Biharmonic conformal immersions into 3-dimensional manifolds

Abstract: Motivated by the beautiful theory and the rich applications of harmonic conformal immersions and conformal immersions of constant mean curvature (CMC) surfaces, we study biharmonic conformal immersions of surfaces into a generic 3-manifold. We first derive an invariant equation for such immersions, we then try to answer the question, "what surfaces can be biharmonically conformally immersed into Euclidean 3-space?" We prove that a circular cylinder is the only CMC surface that can be biharmonically conformally immersed into a Euclidean 3-space; We obtain a classification of biharmonic conformal immersions of complete CMC surfaces into a Euclidean 3-space and a hyperbolic 3-spaces. We also study rotational surfaces that can be biharmonically conformally immersed into a Euclidean 3-space and prove that a circular cone can never such an immersion.