On an energy function for the Weil-Petersson metric on Teichmüller space
Let M be a \(C^{\infty}\) oriented compact manifold without boundary and let \(M_{-1}\) be the tame Fréchet manifold of Riemannian metrics of constant negative curvature. Fix \(g_ 0\in M_{-1}\). If \(g\in M_{-1}\) and \(s: (M,g)\to (M,g_ 0)\) is a smooth \(C^ 1\) map homotopic to the identity, put \(E_ g(s)=\int_{M}(tr ds^*ds)d\mu_ g\) (where \(d\mu_ g\) is the volume element induced by g and the given orientation). Denote by \({\mathcal D}_ 0\) the Lie group of diffeomorphisms of M homotopic to the identity acting on M by pullback. If [g] is the class of g in \(M_{-1}/{\mathcal D}_ 0\), define \(\tilde E[g]=E_ g(s(g))\), where s is the unique harmonic map \((M,g)\to (M,g_ 0)\) homotopic to the identity. The author shows that \([g_ 0]\) is the only critical point of \(\tilde E\) and that the Hessian of \(\tilde E\) at \([g_ 0]\) is (up to a constant) the Weil-Petersson metric.
scientific article; zbMATH DE number 4034693
scientific article