Duke Mathematical Journal

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

Young-Heon Kim

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Let Σ 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 χ(Σ)<0, we show that on Σ, the set of flat metrics that have the same Laplacian spectrum of the Dirichlet boundary condition is compact in the C-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 Σ. Second, we show that the space of such metrics is homeomorphic (in the C-topology) to the space of flat metrics (on Σ) 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 Σ, 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 Σ is of type (g,n), g>0, while Osgood, Phillips, and Sarnak [OPS3] showed the properness when g=0

Article information

Duke Math. J., Volume 144, Number 1 (2008), 73-107.

First available in Project Euclid: 2 July 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58J53: Isospectrality 32G15: Moduli of Riemann surfaces, Teichmüller theory [See also 14H15, 30Fxx]


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

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