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

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- by Christopher M. Judge
- J. Amer. Math. Soc.
**8**(1995), 715-759 - DOI: https://doi.org/10.1090/S0894-0347-1995-1273415-6
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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.## References

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## Bibliographic Information

- © Copyright 1995 American Mathematical Society
- Journal: J. Amer. Math. Soc.
**8**(1995), 715-759 - MSC: Primary 11F72; Secondary 58G25
- DOI: https://doi.org/10.1090/S0894-0347-1995-1273415-6
- MathSciNet review: 1273415