Mathematical Research Letters

Volume 22 (2015)

Number 3

Critical exponent and bottom of the spectrum in pinched negative curvature

Pages: 929 – 944



Thomas Roblin (LPMA / UMR 7599, Université Pierre et Marie Curie, Paris, France)

Samuel Tapie (Laboratoire Jean Leray, Université de Nantes, France)


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.

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