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

Young-Heon Kim
2008 Duke mathematical journal  
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 χ(Σ) < 0, we show that on Σ, the set of flat metrics which have the same Laplacian spectrum of Dirichlet boundary condition is compact in the C ∞ topology. This isospectral compactness extends the result of Osgood, Phillips, and Sarnak [O-P-S3] for type (0, n) surfaces, whose examples include bounded plane domains. Our main ingredients are as following. We first
more » ... that the determinant of the Laplacian is a proper function on the moduli space of geodesically bordered hyperbolic metrics on Σ. Secondly, 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 [O-P-S3] showed the properness when g = 0.
doi:10.1215/00127094-2008-032 fatcat:xegxbgi77vcovhgjfi3pkzxcze