Seminar Outline
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چکیده
1. Teichm uller space and the Teichm uller metric d(X; Y) = inf(1=2) log K(), where : X ! Y ranges over all quasiconformal mappings in the appropriate isotopy class. 2. Teichm uller's theorem: If X is compact, or more generally if X is obtained from a compact Riemann surface by removing a nite number of points, then there is an essentially map f minimizing the innmum above. (This map is unique up to conformal automorphisms of X and Y which are isotopic to the identity). The unique map is characterized by the fact that its Beltrami coeecient = @f=@f is of the form = t=jj, where jtj < 1 and is a holomorphic quadratic diierential on X with R X jj < 1. 3. Conversely, any integrable holomorphic quadratic diierential determines a Teichm uller disk or complex geodesic, by considering the complex structures determined by t=jj for all complex t with jtj < 1. This set of Riemann surfaces X t gives a disk in Teichm uller space isometric to the hyperbolic plane. 4. Another way to look at a Teichm uller disk is in terms of the metric determined by a quadratic diierential. A holomorphic quadratic diieren-tial determines an area form j(z)jjdzj 2 and hence a metric j(z)j 1=2 jdzj. This is nothing more than the Euclidean metric pulled back from the plane by the map obtained by integrating the 1-form given locally by q (z)dz. Since the square-root is determined up to sign, the metric is well-deened, although it has singularities at the zeros of. The underlying conformal structure for this metric is of course the original one. But also determines a family of metrics by considering an aane change of coordinates. Exercise: write down a formula for this family of metrics. 5. Quadratic diierentials, polygons and billiards. A quadratic diierential determines a at metric (with standard singularities) and a foliation 1
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تاریخ انتشار 1993