On the global real analytic coordinates for Teichmüller spaces
Let \(T^{(g,m)}\) denote the Teichmüller space of compact Riemann surfaces of genus g with m boundary components. The geodesic length functions give a natural parametrization for \(T^{(g,m)}\). This means that a Riemann surface of type (g,m) is determined by the lengths of closed geodesic curves of the surface. This paper studies the problem of finding a minimal set of closed curves whose lengths determine the Riemann surface in question. If \(m>0\) then such a minimal set can be found. That is a well known result [cf. e.g. \textit{T. Sorvali}, Ann. Acad. Sci. Fenn., Ser. AI 579, 1-12 (1974; Zbl 0287.30019)]. This paper provides a new proof for this fact. The case \(m=0\) and \(g>1\) is more difficult. It is known that one needs at least 6g-5 geodesic length functions to parametrize the Teichmüller space. Mika Seppälä and Tuomas Sorvali have constructed a set of 6g-4 geodesic length functions which parametrize the Teichmüller space [\textit{M. Seppälä} and \textit{T. Sorvali}, Commet. Math. Helv. 61, 149-160 (1986; Zbl 0606.30042)]. In this paper another such set is constructed.
scientific article; zbMATH DE number 4146364