Biharmonic maps in two dimensions

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 $({\mathbb{R}}^2,{\sigma }^{2}dwd\bar{w})$ 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.

20 pages

References in corpus (5)

• A short survey on biharmonic maps between Riemannian manifolds
• Biharmonic Riemannian submersions from 3-manifolds
• Classifications and Isolation Phenomena of Bi-Harmonic Maps and Bi-Yang-Mills Fields
• On the generalized Chen's conjecture on biharmonic submanifolds
• Some constructions of biharmonic maps and Chen's conjecture on biharmonic hypersurfaces