Biharmonic Conformal Immersions into a 3-Dimensional Conformally Flat Space

This paper investigates biharmonic conformal immersions of surfaces into a conformally flat 3-space. We first establish a characterization of such immersions of totally umbilical surfaces into a generic 3-manifold. It is then proved that any biharmonic conformal immersion of a totally umbilical surface into a nonpositively curved 3-manifold is necessarily a conformal minimal immersion. We further examine the biharmonicity of conformal immersions of totally umbilical planes into a conformally flat 3-space and construct explicit examples of such immersions from a 2-sphere (minus a point) into a conformally flat 3-sphere. Finally, the study is extended to biharmonic conformal immersions of Hopf cylinders associated with a Riemannian submersion.