Biharmonic maps in two dimensions

TL;DR

This paper investigates biharmonic maps between surfaces, providing explicit computations, classifications, and constructions of such maps under various geometric conditions, including conformal and warped product metrics.

Contribution

It offers new formulas for the bitension field of maps between surfaces and classifies linear biharmonic maps in several geometric settings, including spheres and hyperbolic planes.

Findings

Linear maps from Euclidean plane are biharmonic if the conformal factor is bi-analytic.

Classification of linear biharmonic maps between 2-spheres.

Construction of proper biharmonic maps into cones and helicoids.

Abstract

Biharmonic maps between surfaces are studied in this paper. We compute the bitension field of a map between surfaces with conformal metrics in complex coordinates. As applications, we show that a linear map from Euclidean plane into $(\mathbb{R}^2, \sigma^2dwd\bar w)$ is always biharmonic if the conformal factor $\sigma$ is bi-analytic; we construct a family of such $ \sigma$, and we give a classification of linear biharmonic maps between $2$-spheres. We also study biharmonic maps between surfaces with warped product metrics. This includes a classification of linear biharmonic maps between hyperbolic planes and some constructions of many proper biharmonic maps into a circular cone or a helicoid.