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On the existence of Maass cusp forms on hyperbolic surfaces with cone points. (English) Zbl 0846.11035

Let \(\Gamma\) be a non-cocompact cofinite discrete subgroup of \(SL_2 (\mathbb{R} )\) acting on the upper half-plane \(\mathbb{H}\). The Roelcke-Selberg conjecture claims that there exist infinitely many eigenvalues of the Laplacian \(\Delta\) on \(L^2 (\Gamma \backslash \mathbb{H})\). This is well known to hold for congruence subgroups of \(SL_2 (\mathbb{Z})\), but the theoretical and numerical evidence collected to date appears to support the Phillips-Sarnak conjecture which claims that the generic surface \(\Gamma \backslash \mathbb{H}\) has only finitely many (Maaß) cusp forms. Yet, strangely enough, to date no one has found a particular \(\Gamma\) such that the spectral asymptotics for \(\Gamma\) is not the classical one.
A large part of the paper under review is a study of the variational behaviour of the spectrum of the Laplacian under metric perturbation. The main analytical tool is a pseudo-Laplacian in the context of conically singular hyperbolic metrics (defined by the author). In section 5, the monotonicity of eigenvalue branches of the conical pseudo-Laplacian is shown. In particular, the odd eigenvalues for the Hecke groups are uniformly bounded from below by \({1 \over 2} \pi^2\). (This improves on an old but badly known result of W. Roelcke [S.-Ber. Heidelberger Akad. Wiss., Math.-Naturw. Kl. 1953/55, No. 4 (1956; Zbl 0072.30101)] who gave the lower bound \({1 \over 4}\).)
The main disappearance theorem is Theorem 6.1: Let \({\mathcal T}_{g,n,m} \) be the Teichmüller space of hyperbolic metrics on a genus \(g\) surface with \(n\) conical singularities and \(m\) cusps. If there exists a metric in \({\mathcal T}_{g,n - 1,m}\) whose Laplacian has only simple embedded eigenvalues, then the Laplacian of the generic point in \({\mathcal T}_{g,n, m}\) has no square-integrable eigenfunctions with eigenvalue greater than \({1 \over 4}\). In particular, the Laplacian of the generic point has only finitely many square-integrable eigenfunctions. (This result neither contradicts the Roelcke-Selberg conjecture nor affirms the Phillips-Sarnak conjecture.) Moreover, assuming a certain hypothesis, the author shows that at most a countable number of metrics in the Hecke triangle deformation have even cusp forms associated to them. – The Fourier expansion of an eigenfunction in the model hyperbolic cone and its asymptotics near the cone point are determined in an appendix.

MSC:

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
58J50 Spectral problems; spectral geometry; scattering theory on manifolds

Citations:

Zbl 0072.30101
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References:

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