Homotopy of curves and mappings and the Teichmüller metric
Let \(T\) be the Teichmüller space consisting of equivalence classes of finite bordered Riemann surfaces of given genus and connectivity with a given finite number of functions with a topological determination (``marked''). For the model surface \((S_0,id)\) and \((S,f)\) the family \({\mathcal F}\) of homotopy classes \(\Gamma\) determined by Jordan curves on \(S_0\) not homotopic to a point determines a relationship to the corresponding \(f(\Gamma)\) on \(S\). Let \(m(\Gamma), m(f(\Gamma))\) be the appropriate modules. The author considers the quantity \[ m_\Gamma(0,x)= \underset {\Gamma\in{\mathcal F}} {\text{l.u.b.}} \frac12 \Biggl|\log \frac{m(f(\Gamma))} {m(\Gamma)} \Biggr| \] where \(0,x\) denote the elements of \(T\) determined by \((S_0,id)\) and \((S,f)\). It is not hard to see that this defines a distance from 0 to \(x\). Defining the corresponding quantity \(m(x,x')\) for \(x,x'\) determined by \((S,f)\), \((S',f')\) one gets a metric on \(T\). It is not difficult to see that this is equivalent to the Teichmüller metric.
scientific article; zbMATH DE number 1037782