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On the Laplacian and the geometry of hyperbolic 3-manifolds. (English) Zbl 0763.53040
Let \(N=\mathbb{H}^ 3/\Gamma\) be an infinite volume hyperbolic 3-manifold which is homeomorphic to the interior of a compact manifold. We prove that if \(N\) is not geometrically finite, then \(\lambda_ 0(N)=0\) and if \(N\) is geometrically finite we produce an upper bound for \(\lambda_ 0(N)\) in terms of the volume of the convex core. As a consequence we see that \(\lambda_ 0(N)=0\) if and only if \(N\) is not geometrically finite. We also show that if \(N\) has a lower bound for its injectivity radius and is not geometrically finite, then its limit set \(L_ \Gamma\) has Hausdorff dimension 2.

MSC:
53C20 Global Riemannian geometry, including pinching
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
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