Biharmonic Maps Between Surfaces

In this thesis, we study biharmonic maps between Surfaces (i.e.,2-dimensional Rie-mannian manifolds). First, we obtain the equation of such maps in complex form by using isothermal coordinates. This gives a generalization of the well-known equation of harmonic maps in complex form. Then, as applications, we study the linear biharmonic maps between constant Gaussian curvature surfaces and discover a family of conformal metrics in the target surface such that any linear map from Euclidean plane into this target surface is a biharmonic map. We also study the biharmonic maps between surfaces with warped product metrics and study the problem of biharmonic metrics, that is, under what conditions a warped product metric is biharmonic with respect to a Euclidean metric or a hyperbolic metric.