GRAFTING, HARMONIC MAPS AND PROJECTIVE-STRUCTURES ON SURFACES

Grafting is a surgery on Riemann surfaces introduced by Thurston; it c onnects hyperbolic geometry and the theory of projective structures on surfaces. ([4], [7]) We will discuss the space of projective structur es in terms of the Thurston's geometric parametrization given by graft ing, From this approach we will prove that on any compact Riemann surf ace with genus greater than 1 there exist infinitely many projective s tructures with Fuchsian holonomy representations. In course of the pro of it will turn out that grafting is closely related to harmonic maps between surfaces.