Timelike surfaces in the Lorentz–Minkowski space with prescribed Gaussian curvature and Gauss map

In this paper we obtain a Kenmotsu-type representation for timelike surfaces with prescribed Gauss map in the three-dimensional Lorentz–Minkowski space L3. We use such representation in order to classify the complete timelike surfaces with positive constant Gaussian curvature in L3 in terms of harmonic diffeomorphisms between simply connected Lorentz surfaces and the universal covering of the de Sitter Space.