Surfaces with boundary: Their uniformizations, determinants of Laplacians, and isospectrality

Abstract

Let $\Sigma$ be a compact surface of type $(g,n)$, $n>0$, obtained by removing $n$ disjoint disks from a closed surface of genus $g$. Assuming that $\chi (\Sigma )<0$, we show that on $\Sigma$, the set of flat metrics that have the same Laplacian spectrum of the Dirichlet boundary condition is compact in the $C^{\infty }$-topology. This isospectral compactness extends the result of Osgood, Phillips, and Sarnak [OPS3, Theorem 2] for surfaces of type $(0,n)$ whose examples include bounded plane domains.

Our main ingredients are as follows. We first show that the determinant of the Laplacian is a proper function on the moduli space of geodesically bordered hyperbolic metrics on $\Sigma$. Second, we show that the space of such metrics is homeomorphic (in the $C^{\infty }$-topology) to the space of flat metrics (on $\Sigma$) with constantly curved boundary. Because of this, we next reduce the complicated degenerations of flat metrics to the simpler and well-known degenerations of hyperbolic metrics, and we show that determinants of Laplacians of flat metrics on $\Sigma$, with fixed area and boundary of constant geodesic curvature, give a proper function on the corresponding moduli space. This is interesting because Khuri [Kh] showed that if the boundary length (instead of the area) is fixed, the determinant is not a proper function when $\Sigma$ is of type $(g,n), g>0$, while Osgood, Phillips, and Sarnak [OPS3] showed the properness when $g=0$