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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

On the existence of Maass cusp forms on hyperbolic surfaces with cone points


Author: Christopher M. Judge
Journal: J. Amer. Math. Soc. 8 (1995), 715-759
MSC: Primary 11F72; Secondary 58G25
MathSciNet review: 1273415
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Abstract: The perturbation theory of the Laplace spectrum of hyperbolic surfaces with conical singularities belonging to a fixed conformal class is developed. As an application, it is shown that the generic such surface with cusps has no Maass cusp forms ($ {L^2}$ eigenfunctions) under specific eigenvalue multiplicity assumptions. It is also shown that eigenvalues depend monotonically on the cone angles. From this, one obtains Neumann eigenvalue monotonicity for geodesic triangles in $ {{\mathbf{H}}^2}$ and a lower bound of $ \frac{1}{2}{\pi ^2}$ for the eigenvalues of `odd' Maass cusp forms associated to Hecke triangle groups.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0894-0347-1995-1273415-6
PII: S 0894-0347(1995)1273415-6
Keywords: Maass cusp form, embedded eigenvalues, Hecke triangle group, hyperbolic surfaces
Article copyright: © Copyright 1995 American Mathematical Society