Heights on the moduli space of Riemann surfaces with circle boundaries (Q1190831)

Let \(\Sigma=\Sigma_{p,n}\) be a compact Riemann surface of genus \(p\) with \(n\) distinct open disks removed. Associated with a smooth metric \(\sigma\) on \(\Sigma\) is the Beltrami-Laplace operator \(\Delta=\Delta_ \sigma\) with Dirichlet boundary conditions. The height function \(h(\sigma)\) is a generalization of the classical Dirichlet integral. This function is defined as \(h(\sigma)=-\log(\text{det} \Delta)\) where \(\text{det} \Delta=\exp(-Z'(0))\) and \(Z(s)=\sum_{j=1}^ \infty \lambda_ j^{-s}\) is a zeta function built up from the eigenvalues \(\lambda_ j\) of \(\Delta\). Let \(M_ u(\Sigma)\) be the set of flat metrics on \(\Sigma\) with geodesic curvature \(-1\) on \(\partial\Sigma\). The author proves --- if \(pn\neq 0\) --- that \(h: M_ u(\Sigma)\to\mathbb{R}\) is not a proper map and discusses the boundary behavior of this function.