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Surfaces with boundary: their uniformizations, determinants of Laplacians, and isospectrality. (English) Zbl 1146.58027

Summary: Let \(\Sigma\) 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 \(\chi(\Sigma)<0\), we show that on \(\Sigma\), the set of flat metrics that have the same Laplacian spectrum of the Dirichlet boundary condition is compact in the \(C^\infty\)-topology. This isospectral compactness extends the result of B. Osgood, R. Phillips, and S. Sarnak [Ann. Math. (2) 129, No. 2, 293–362 (1989; Zbl 0677.58045), 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 \(\Sigma\). Second, we show that the space of such metrics is homeomorphic (in the \(C^\infty\)-topology) to the space of flat metrics (on \(\Sigma\)) 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 \(\Sigma\), with fixed area and boundary of constant geodesic curvature, give a proper function on the corresponding moduli space. This is interesting because H. H. Khuri [Duke Math. J. 64, No. 3, 555–570 (1991; Zbl 0755.30037] showed that if the boundary length (instead of the area) is fixed, the determinant is not a proper function when \(\Sigma\) is of type \((g,n)\), \(g>0\), while Osgood, Phillips, and Sarnak [loc. cit.] showed the properness when \(g=0\).

MSC:

58J53 Isospectrality
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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