Biharmonic conformal immersions into a 3-dimensional conformally flat space

TL;DR

This paper investigates biharmonic conformal immersions of surfaces into conformally flat 3-spaces, providing characterizations, classification methods, and numerous explicit examples, including immersions of spheres and Hopf cylinders.

Contribution

It offers a new characterization and classification approach for biharmonic conformal immersions into conformally flat 3-spaces, with explicit constructions and examples.

Findings

Characterization of biharmonic conformal immersions of totally umbilical surfaces.

Method for constructing biharmonic conformal immersions into conformally flat 3-spaces.

Explicit examples including immersions of spheres and Hopf cylinders.

Abstract

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 $S^2(\frac{1}{\sqrt{2}})\to S^3$ which was found in [2]. Finally, we study biharmonic conformal immersions of Hopf cylinders of a Riemannian submersion.