Biharmonic maps in two dimensions

Details Biharmonic maps in two dimensions Biharmonic maps in two dimensions

2010-08-04

arxiv.org/abs/1008.0819v1

Mathematics

Differential Geometry

20 pages

Scientific paper

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.

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