CONFORMAL GEOMETRY SEMINAR The Poincaré Uniformization Theorem

We will assume that all the manifolds M are compact and orientable unless otherwise stated. In this first part of the seminar we will prove the Poincaré Uniformization Theorem. Theorem 1 (Poincaré Uniformization Theorem). Let (M, g) be a compact 2-dimensional Riemannian manifold. Then there is a metric �g = e 2u g conformal to g which has constant Gauss curvature constant. 1. Preliminaries 1.1. Geometry. A covariant derivative on a manifold M is an operator ∇XY on vector fields X and Y satisfying for any smooth function f: (i) ∇fXY = f∇XY; and (ii) ∇X(fY) = f∇XY + (∇Xf)Y. If g is a Riemannian metric on M, then there is associated with � g a � unique covariant derivative ∇ characterized by: (iii) ∇XY − ∇Y X = [X, Y]; and (iv) ∇X g(Y, Z) = g(∇XY, Z) + g(Y, ∇XZ). We define the Christoffel symbols by Γi jk = dxi�∇∂j ∂k, where ∂i is a coordinate basis, and dxi is the dual basis. The Christoffel symbols can be computed from: Γ i 1 jk = 2 gim (∂jgmk + ∂kgmj − ∂mgjk). A curve γ is a geodesic if ∇ ˙γ ˙γ = 0. Geodesics locally minimize arclength � γ | ˙γ|. A Riemannian manifold is complete if there are no inextendible geodesics. In a complete Riemannian manifold, any two points can be joined by a length minimizing geodesic. If M is a complete Riemannian manifold, and x ∈ M, the map TxM → M sending each X ∈ TxM to the point γ(1), where γ is the geodesic with γ(0) = x and ˙γ(0) = X, is denoted expx, and is called the exponential map at x. The radius of injectivity ix at x is the supremum over all R> 0 such that expx is non-singular on BR(0) ⊂ TxM. The Riemann curvature operator is defined by