Harmonic properties of the module of a family of curves and invariant metrics in Teichmüller space
Let \(T(g,n)\) be the Teichmüller space of equivalence classes of finite bordered Riemann surfaces of genus \(g\) and connectivity \(n\) with \(3g+3>n\). On this are defined various metrics, in particular the biholomorphically invariant Carathéodory metric \(c_T\) and the Kobayashi metric \(k_T\). The author obtains a sufficient condition for local coincidence of these metrics utilizing the reviewer's fundamental theorem on the modules of curve families. This is used to define a functional \(m(x)\) on \(T(g,n)\) and the result is as follows (where \(m_0\)) is the module for the model surface). Let \(\varphi\) be the lifting to \(U=\{|z|<1\}\) of the extremal quadratic differential for \(m_0\) and let \(m(x)\) be a pluriharmonic function on \(T(g,n)\). Then the coincidence of the metrics \(c_T\) and \(k_T\) holds in the geodesic Teichmüller disc \(\Phi(t \overline{\varphi}) |\varphi|)\), \(t\in U\). The author's reference to the reviewer's paper [7] is incorrect. The journal is Ann. Math. The data is 1957 not 1975.
scientific article; zbMATH DE number 1050640