Surfaces with boundary: their uniformizations, determinants of Laplacians, and isospectrality
| dc.creator | Kim, Young-Heon | |
| dc.date | 2006-09-04 | |
| dc.date | 2007-06-04 | |
| dc.date.accessioned | 2026-07-07T08:08:08Z | |
| dc.date.available | 2026-07-07T08:08:08Z | |
| dc.description | 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^\infty topology. This isospectral compactness extends the result of Osgood, Phillips, and Sarnak \cite{OPS3} 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^\infty-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 \cite{Kh} 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 \cite{OPS3} showed the properness when g=0. | |
| dc.description | 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 | |
| dc.identifier | https://arxiv.org/abs/math/0609085 | |
| dc.identifier | http://arxiv.org/abs/math/0609085 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/131162 | |
| dc.subject | Differential Geometry | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 58J53; 32G15 | |
| dc.title | Surfaces with boundary: their uniformizations, determinants of Laplacians, and isospectrality | |
| dc.type | text |