## Duke Mathematical Journal

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

Young-Heon Kim

#### Abstract

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 $\chi(\Sigma) \lt 0$, 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 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 $\Sigma$. 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 Khuri [Kh] 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 [OPS3] showed the properness when $g=0$

#### Article information

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

Dates
First available in Project Euclid: 2 July 2008

https://projecteuclid.org/euclid.dmj/1215032811

Digital Object Identifier
doi:10.1215/00127094-2008-032

Mathematical Reviews number (MathSciNet)
MR2429322

Zentralblatt MATH identifier
1146.58027

#### Citation

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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