Non-convexity of spheres in infinite-dimensional Teichmüller spaces
The Teichmüller space, \(T(G)\), of a Fuchsian group \(G\), is infinite- dimensional if either \(G\) is of the second kind or it is infinitely generated (or both). The Teichmüller metric is always the Kobayashi metric of \(T(G)\). Let \(S_ r\) denote the sphere of points at a fixed distance \(r\) from a given point in \(T(G)\). It is shown that there exist two points on \(S_ r\), and a Teichmüller geodesic joining them, such that the entire geodesic arc lies on that same \(S_ r\). The method is to construct a special holomorphic quadratic differential for \(G\), and thereby certain extremal Beltrami differentials. The same quadratic differential has been used by the author in another paper (``Closed geodesics and non-differentiability of the metric in infinite-dimensional Teichmüller spaces'', Proc. Am. Math. Soc., to appear) to prove stronger results than the above non-convexity.
scientific article; zbMATH DE number 682135