Asymptotic behavior of, small eigenvalues, short geodesics and period matrices on degenerating hyperbolic Riemann surfaces

Grotowski, J. F., Huntley, J. and Jorgenson, J. (2001) Asymptotic behavior of, small eigenvalues, short geodesics and period matrices on degenerating hyperbolic Riemann surfaces. Forum Mathematicum, 13 6: 729-740. doi:10.1515/form.2001.031

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Author Grotowski, J. F.
Huntley, J.
Jorgenson, J.
Title Asymptotic behavior of, small eigenvalues, short geodesics and period matrices on degenerating hyperbolic Riemann surfaces
Journal name Forum Mathematicum   Check publisher's open access policy
ISSN 0933-7741
1435-5337
Publication date 2001
Sub-type Article (original research)
DOI 10.1515/form.2001.031
Open Access Status File (Publisher version)
Volume 13
Issue 6
Start page 729
End page 740
Total pages 12
Place of publication Berlin, Germany
Publisher Walter de Gruyter GmbH
Language eng
Abstract Consider {M-t} a semi-stable family of compact, connected algebraic curves which degenerate to a stable, noded curve M-0. The uniformization theorem allows us to endow each curve M-t in the family, as well as the limit curve M-0 (after its nodes have been removed), with its natural complete hyperbolic metric (i.e. constant negative curvature equal to -1), so that we are considering a degenerating family of compact hyperbolic Riemann surfaces. Assume that M-0 has k components and n nodes, so there are n families of geodesics whose lengths approach zero under degeneration and k - 1 families of eigenvalues of the Laplacian which approach zero under degeneration. A problem which has received considerable attention is to compare the rate at which the eigenvalues and the lengths of geodesics approach zero. In this paper, we will use results from complex algebraic geometry and from heat kernel analysis to obtain a precise relation involving the small eigenvalues, the short geodesics, and the period matrix of the underlying complex curve M-t. Our method leads naturally to a general conjecture in the setting of an arbitrary degenerating family of hyperbolic Riemann surfaces of finite volume.
Keyword Mathematics, applied
Mathematics
Geometry
Space
Q-Index Code C1

Document type: Journal Article
Sub-type: Article (original research)
Collection: School of Physical Sciences Publications
 
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Created: Wed, 17 Oct 2007, 11:00:30 EST