Complex projective structures, grafting, and Teichmuller theory

We study the space P (S) of marked complex projective ( CP1 ) structures on a compact surface in terms of Teichmuller theory and hyperbolic geometry. In particular, we show that the structure of this space as a bundle over the Teichmuller space T (S) of conformal structures is compatible with Thurston's parameterization of P (S) using grafting, and we provide an explicit description of the boundary of the fiber P(X) over X ∈ T (S) in terms of an involution on the space PML (S) of projective measured laminations. This involution encodes the conformal geometry of X via the orthogonality of the vertical and horizontal measured foliations of holomorphic quadratic differentials.; We also apply these results to study the projective structures with Fuchsian holonomy on a fixed Riemann surface X. We formulate a general conjecture comparing these Fuchsian centers and associated Strebel differentials on X, and then prove that this conjecture holds along rays in ML (S) supported on finitely many simple closed curves. This generalizes previous results about Fuchsian centers obtained by Anderson.; The proofs of these results involve the theory of harmonic maps between Riemann surfaces and from Riemann surfaces to R -trees. Specifically, we show that the canonical collapsing and co-collapsing maps associated to a complex projective surface are nearly harmonic, and then apply existing results about harmonic maps and their Hopf differentials.