  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 \chi(\Sigma)<0, we
show that on \Sigma, the set of flat metrics which have the same Laplacian
spectrum of Dirichlet boundary condition is compact in the C^\infty topology.
This isospectral compactness extends the result of Osgood, Phillips, and Sarnak
\cite{OPS3} for type (0,n) surfaces, whose examples include bounded plane
domains.
  Our main ingredients are as following. We first show that the determinant of
the Laplacian is a proper function on the moduli space of geodesically bordered
hyperbolic metrics on \Sigma. Secondly, 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 \cite{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 \cite{OPS3}
showed the properness when g=0.
