9g-9-Theorem - 9g-9-Theorem

In math , that is-Theorem a theorem from the Teichmüller theory .

It says that it is on a ( closed , orientable ) area of the sex a lot of closed curves whose lengths uniquely define each hyperbolic metric on the surface.

The sentence thus embeds the Teichmüller space in thewhich, however, is not surjective . A diffeomorphism of the Teichmüller space withis instead realized by the Fennel-Nielsen coordinates , which are a selected set of the length and the twist parameters of the corresponding geodesics are assigned to closed curves.

The sentence is generalized to (orientable) areas by gender With and tips where then the lengths of closed geodesics are required.

Hamenstädt has shown that in the case of closed areas even the lengths closed geodesics can determine the hyperbolic metric, while the lengths of Geodesics are not sufficient for this. For surfaces with points you need Geodesics.

literature

  • Benson Farb, Dan Margalit: A primer on mapping class groups. (= Princeton Mathematical Series. 49). Princeton University Press, Princeton, NJ 2012, ISBN 978-0-691-14794-9. (online; pdf)
  • Ursula Hamenstädt: Length functions and parametrization of Teichmüller space for surfaces with cusps. Ann. Acad. Sci. Fenn. Math. 28, 75–88 (2003).