Duke Mathematical Journal
- Duke Math. J.
- Volume 144, Number 1 (2008), 73-107.
Surfaces with boundary: Their uniformizations, determinants of Laplacians, and isospectrality
Let be a compact surface of type , , obtained by removing disjoint disks from a closed surface of genus . Assuming that , we show that on , the set of flat metrics that have the same Laplacian spectrum of the Dirichlet boundary condition is compact in the -topology. This isospectral compactness extends the result of Osgood, Phillips, and Sarnak [OPS3, Theorem 2] for surfaces of type 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 . Second, we show that the space of such metrics is homeomorphic (in the -topology) to the space of flat metrics (on ) 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 , 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 is of type , while Osgood, Phillips, and Sarnak [OPS3] showed the properness when
Duke Math. J., Volume 144, Number 1 (2008), 73-107.
First available in Project Euclid: 2 July 2008
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Kim, Young-Heon. Surfaces with boundary: Their uniformizations, determinants of Laplacians, and isospectrality. Duke Math. J. 144 (2008), no. 1, 73--107. doi:10.1215/00127094-2008-032. https://projecteuclid.org/euclid.dmj/1215032811