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 $\chi(\Sigma) \lt 0$ , 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 $g=0$