2006/09/04 by Young-Heon Kim, Kim, Young-Heon
Mathematics · #32G15 #58J53 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #math.AP #math.DG #msc:32G15 #msc:58J53
paper · pdf · doi:10.48550/arxiv.math/0609085
Further Revised. A technical error is corrected; the sections devoted to the proof of the insertion lemma and the separation of variables method are completely rewritten. (Sections 4, 5, and 6 in this revised version.) A lot of changes, corrections, and improvements are made throughout the paper. No mathematical change in the main theorems listed in the introduction
Let Σbe 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 \citeOPS3 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 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 \citeKh showed that if the boundary length (instead of the area) is fixed, the determinant is not a proper function when Σis of type (g, n), g>0; while Osgood, Phillips, and Sarnak \citeOPS3 showed the properness when g=0.