Mathematical Research Letters
Volume 22 (2015)
Critical exponent and bottom of the spectrum in pinched negative curvature
Pages: 929 – 944
In this note, we present a new proof of the celebrated theorem of Patterson–Sullivan which relates the critical exponent of a hyperbolic manifold and the bottom of its spectrum. The proof extends to manifolds with pinched negative curvatures. It provides a sufficient criterion for the existence of isolated eigenvalues for the Laplacian on geometrically finite manifolds with pinched negative curvatures.