Pignataro, Thea; Sullivan, Dennis Ground state and lowest eigenvalue of the Laplacian for non-compact hyperbolic surfaces. (English) Zbl 0602.58046 Commun. Math. Phys. 104, 529-535 (1986). Let M be a complete Riemannian surface with constant curvature -1 and a finitely generated fundamental group. The authors estimate the first eigenvalue of the Laplacian giving upper and lower bounds, and they give information on the shape of the basic eigenfunction. Reviewer: U.Simon Cited in 4 Documents MSC: 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 58J60 Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) 53C20 Global Riemannian geometry, including pinching Keywords:first eigenvalue of the Laplacian; first eigenfunction of the Laplacian; complete Riemann surface PDFBibTeX XMLCite \textit{T. Pignataro} and \textit{D. Sullivan}, Commun. Math. Phys. 104, 529--535 (1986; Zbl 0602.58046) Full Text: DOI References: [1] Patterson, S.J.: The limit set of a Fuchsian group. Acta Math.136, 241-273 (1976) · Zbl 0336.30005 · doi:10.1007/BF02392046 [2] Pignataro, T.: Hausdorff dimension, spectral theory and applications to the quantization of geodesic flows on surfaces of constant negative curvature. Thesis, Princeton University (1984), Sects. 3.2 and 4.4 [3] Sullivan, D.: Entropy, Hausdorff measures old and new and the limit sets of geometrically finite Kleinian groups. Acta Math.153, 259 (1984) · Zbl 0566.58022 · doi:10.1007/BF02392379 [4] Sullivan, D.: Related aspects of positivity, ground states, Hausdorff dimension, complementary series, ..., preprint IHES 1983 [5] Thurston, W.: Topology and geometry of three-manifolds. Princeton University 1979 · Zbl 0409.58001 [6] Schoen, R., Wolpert, S., Yau, S.-T.: Geometric bounds on the low eigenvalues of a compact surface. AMS Proc. Symp. Pure Math. Vol.36 (1980) · Zbl 0446.58018 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.