A Short Course on Teichmüller’s Theorem

We present a brief exposition of Teichmüller’s theorem. Introduction An orientation preserving homeomorphism f from a Riemann surface X onto a Riemann surface Y is given. Teichmüller’s problem is to find a quasiconformal homeomorhism in the homotopy class of f with minimal maximal dilatation, that is, to find a homeomorphism f0 whose maximal dilatation K(f0) is as small as possible in its homotopy class. Teichmüller’s theorem states that the problem has a unique extremal solution provided that X is compact or compact except for a finite number of punctures, namely, a Riemann surface of finite analytic type. Moreover, except when f0 is conformal, f0 is equal to a stretch mapping along the horizontal trajectories of some uniquely determined holomorphic quadratic differential φ(z)(dz), with ∫ ∫ X |φ|dxdy = 1, postcomposed by a conformal map. It turns out that even for arbitrary Riemann surfaces, whether or not they are of finite analytic type, this statement is generically true (see [20], [27]). The goal of this course is to present a brief proof of the original Teichmüller theorem in a series of lectures and exercises on the following topics: 1. conformal maps and Riemann surfaces, 2. quasiconformal maps, dilatation and Beltrami coefficients, 3. extremal length, 4. the Beltrami equation, 5. the Reich-Strebel inequality and Teichmüller’s uniqueness theorem, 6. the minimum norm principle, 7. the heights argument, 8. the Hamilton-Krushkal condition and Teichmüller’s existence theorem,