15 July 2008 Surfaces with boundary: Their uniformizations, determinants of Laplacians, and isospectrality
Young-Heon Kim
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Duke Math. J. 144(1): 73-107 (15 July 2008). DOI: 10.1215/00127094-2008-032


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


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Young-Heon Kim. "Surfaces with boundary: Their uniformizations, determinants of Laplacians, and isospectrality." Duke Math. J. 144 (1) 73 - 107, 15 July 2008. https://doi.org/10.1215/00127094-2008-032


Published: 15 July 2008
First available in Project Euclid: 2 July 2008

zbMATH: 1146.58027
MathSciNet: MR2429322
Digital Object Identifier: 10.1215/00127094-2008-032

Primary: 32G15 , 58J53

Rights: Copyright © 2008 Duke University Press


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Vol.144 • No. 1 • 15 July 2008
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