Projective structures on Riemann surfaces and developing maps to H(3) and CP(n)

We study projective structures on Riemann surfaces through the geometry of their maps to hyperbolic three-space H3 and complex projective n-space CPn. A projective surface is a topological surface X with a coordinate covering { Ua,za } such that the transition functions za&j0; z-1b are Mobius transformations. The hyperbolic metric (or the Kobayashi metric) on X is the conformal metric K = K(z)|dz| which gives the length of a tangent vector v as inf rf*v where the infimum is taken over all holomorphic immersions f : D→X of the unit disk and rho = 2|dz|/(1 -- | z|2) is the hyperbolic metric on Delta. The projective metric T = T(z)|dz| (or the Thurston metric) is given by the same formula with the infimum now taken only over projective immersions. In general K ≤ T, and K depends only on the complex structure. The uniformization theorem provides a canonical hyperbolic structure on most Riemann surfaces. The difference between a given projective structure on X and the hyperbolic structure is measured by a quadratic differential &phis;. The Linfinity norm of &phis; is ∥f ∥infinity sup = |&phis;|/K2. Our main result is the distortion theorem T≤1+2∥ f∥ infinityK. To a projective surface X with a conformal metric rho, we functorially associate a developing map to H3 . Our theorem is proved by comparing the curvatures of the image surfaces corresponding to K and T. We include graphics showing the surface corresponding to K and the projective structure on a half-plane determined by the function f( z) = zalpha for various values of alpha. The inequality gives a new proof of the Koebe 1/4 Theorem. We also apply the inequality to problems suggested by grafting and pleated surfaces in quasifuchsian manifolds. For example, when X arises from Y by grafting along the geodesic gamma with weight alpha, we obtain a2g-2ℓ Xg≤ 1+2∥f ∥infinity , and a2g-2ℓ Yg≤1+2∥ f ∥infinity. Two projective structures on the same Riemann surface differ by a quadratic differential, and we establish a similar relation between equivariant immersions of X&d5; into CPn and sections of T*Xn+1 for n≥1. This relationship gives a geometric interpretation of Bers' Area Theorem.