Biharmonic conformal immersions into a 3-dimensional conformally flat space

Inspired by the work of Ou 12, 17, we study biharmonic conformal immersions of surfaces into a conformally flat 3-space. We first give a characterization of biharmonic conformal immersions of totally umbilical surfaces into a generic 3-manifold. As an application, we give a method to produce biharmonic conformal immersions into a conformally flat 3-space. We then use the method to obtain a classification of biharmonic maps in a family of conformal immersions and construct many examples of biharmonic conformal immersions from a 2-sphere into a conformal 3-sphere. Our examples include proper biharmonic conformal immersions of a 2-sphere minus a point into a conformal 3-sphere with nonconstant conformal factor and the biharmonic isometric immersion S² (12) S³ which was found in 2. Finally, we study biharmonic conformal immersions of Hopf cylinders of a Riemannian submersion.