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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 (R 2 , σ 2 dwdw) is always biharmonic if the conformal factor σ is bianalytic; we construct a family of such σ, 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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