Critical exponent and bottom of the spectrum in pinched negative curvature

Thomas Roblin (1) , Samuel Tapie (2)

1 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
2 LMJL - Laboratoire de Mathématiques Jean Leray

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.

Domaines: Géométrie différentielle [math.DG]; Systèmes dynamiques [math.DS]

Thomas Roblin, Samuel Tapie. Critical exponent and bottom of the spectrum in pinched negative curvature. Mathematical Research Letters, 2015, 22 (3), pp.929 - 944. ⟨10.4310/MRL.2015.v22.n3.a15⟩. ⟨hal-01729692⟩