On conformal biharmonic immersions

TL;DR

This paper investigates conformal biharmonic immersions, deriving invariant equations, constructing explicit examples, and exploring their geometric properties and relationships with holomorphicity.

Contribution

It introduces an invariant equation for conformal biharmonic immersions and provides numerous explicit examples, expanding understanding of proper biharmonic maps.

Findings

Constructed a 2-parameter family of non-minimal conformal biharmonic immersions of cylinders into R^3.

Provided examples of conformal biharmonic immersions from 4D Euclidean space into spheres and hyperbolic spaces.

Suggested the abundance of proper biharmonic maps within conformal immersions.

Abstract

This paper studies conformal biharmonic immersions. We first study the transformations of Jacobi operator and the bitension field under conformal change of metrics. We then obtain an invariant equation for a conformal biharmonic immersion of a surface into Euclidean 3-space. As applications, we construct a 2-parameter family of non-minimal conformal biharmonic immersions of cylinder into $\r^3$ and some examples of conformal biharmonic immersions of 4-dimensional Euclidean space into sphere and hyperbolic space thus provide many simple examples of proper biharmonic maps with rich geometric meanings. These suggest that there are abundant proper biharmonic maps in the family of conformal immersions. We also explore the relationship between biharmonicity and holomorphicity of conformal immersions of surfaces.