2012/09/10 by Ye‐Lin Ou, Ou, Ye-Lin
Mathematics · #58E20 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.1209.2104
openalex publication_date 2012/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.