Asymptotic behavior of quadratic differentials

Abstract

ii Summary Every closed oriented surface S of genus g ≥ 2 can be endowed with a non-unique hyperbolic metric. By the celebrated Uniformization Theorem, hyperbolic and complex structures on S are in one-to-one correspondence. A question that arises is: Can we parametrize the complex structures in a nice way? This question brings us to Teichmüller theory. Teichmüller space is the quotient of all complex structures on S by the group of diffeomorphisms isotopic to the identity. The quotient of the Teichmüller space by the mapping class group, i.e. the group of isotopy classes of orientation preserving diffeomorphisms of S, is the moduli space, i.e. the set of all biholomorphic equivalence classes of complex structures S can be endowed with. The moduli space classifies Riemann surfaces of a given genus up to biholomor-phic equivalence. But it turns out that, instead of looking at the structure of the moduli space directly, it is easier to study the Teichmüller space: The Teichmüller space is a complex manifold biholomorphic to a bounded domain in C 3g−3 , whereas the moduli space is an orbifold rather than a manifold and has complicated topol-ogy. There are several mapping class group-invariant metrics on Teichmüller space which are useful when studying the structure of the moduli space and the Teich-müller space. Among them is the Teichmüller metric, a complete Finsler metric that is well-suited to measure differences in complex structures. For any pair of points in Teichmüller space there is a homeomorphism which maps one point to the other (respecting the marking) and which is quasi-conform with respect to the two complex structures. The Teichmüller metric measures how much the " optimal " quasi-conformal map differs from a biholomorphic isomorphism. The cotangent bundle of the Teichmüller space can be identified with the bundle of holomorphic quadratic differentials. There is a one-to-one correspondence between unit speed Teichmüller geodesics and holomorphic quadratic differentials of unit area. The vertical foliations of quadratic differentials play the role of " directions " in Teichmüller space. We call a pair of geodesics with common direction asymptotic if there are unit speed parametrizations of the geodesics such that the distance between the two geodesics converges to zero. Teichmüller space together with the Teichmüller metric is not negatively curved in any reasonable sense, but it shares many properties with negatively curved spaces. In 1980 Masur proved that for almost all directions pairs of Teichmüller geodesics …

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Cite this paper

@inproceedings{Nipper2010AsymptoticBO, title={Asymptotic behavior of quadratic differentials}, author={Emanuel Josef Nipper and Werner Ballmann}, year={2010} }