Harmonic maps to Teichmüller space
Let \(B\) be a closed, compact Riemann surface, and \(\Gamma_g\) denote the mapping class group for a closed oriented genus \(g>1\) surface \(\Sigma\). A representation \(\rho:\pi_1(B)\to\Gamma_g\) gives a homotopy class of \(\rho\)-equivariant continuous maps \(u:\widetilde B\to T_g\). Here, \(T_g\) is the Teichmüller space of equivalence classes of conformal structures on \(\Sigma\), and \(\widetilde B\) denotes the universal cover of \(B\). Theorem 1.2. Let \(T_g\) be equipped with the McMullen metric \(g_{\frac 1l}\) which is complete \(\Gamma_g\)-invariant Kähler. If \(B\) is a compact surface and the image of \(\rho:\pi_1(B)\to\Gamma_g\) is either finite or sufficiently large, then there exists a \(\rho\)-equivariant harmonic map \(u:\widetilde B\to T_g\). Similary, if \(\rho:\pi_1(B')\to\Gamma_g\) (\(B'=B\smallsetminus\{p_1,p_2,\dots,p_k\}\)) is either finite or sufficiently large and there exists a finite energy \(\rho\)-equivariant map \(f:\widetilde B'\to T_g\), then there exists a finite energy \(\rho\)-equivariant harmonic map \(u:\widetilde B'\to T_g\). Theorem 1.3. Let \(X\to B\) be any symplectic Lefschetz fibration of genus \(>1\) over \(B=S^2\) with monodromy \(\rho\). Then there exists a finite energy \(\rho\)-equivariant harmonic map \(u:\widetilde B'\to T_g\), where \(T_g\) again has the metric \(g_{\frac 1l}\).
scientific article; zbMATH DE number 1489541