Resolvent of the Laplacian on geometrically finite hyperbolic manifolds

Colin Guillarmou (1) , Rafe Mazzeo (2)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 Stanford University

For geometrically finite hyperbolic manifolds $\Gamma\backslash H^{n+1}$, we prove the meromorphic extension of the resolvent of Laplacian, Poincaré series, Einsenstein series and scattering operator to the whole complex plane. We also deduce the asymptotics of lattice points of $\Gamma$ in large balls of $H^{n+1}$ in terms of the Hausdorff dimension of the limit set of $\Gamma$.

Domaines: Théorie spectrale [math.SP]; Géométrie différentielle [math.DG]

Colin Guillarmou, Rafe Mazzeo. Resolvent of the Laplacian on geometrically finite hyperbolic manifolds. Inventiones Mathematicae, 2012, 187 (1), pp.99--144. ⟨hal-00455589v2⟩