Grafting, harmonic maps and projective structureson surfaces

Abstract

Grafting is a surgery on Riemann surfaces introduced by Thurston; it connects hyperbolic geometry and the theory of projective structures on surfaces. ([4], [7]) We will discuss the space of projective structures in terms of the Thurston’s geometric parametrization given by grafting. From this approach we will prove that on any compact Riemann surface with genus greater than 1 there exist infinitely many projective structures with Fuchsian holonomy representations. In course of the proof it will turn out that grafting is closely related to harmonic maps between surfaces. © 1997 Journal of Differential Geometry. © 1997 Applied Probability Trust.