Spectral gap for surfaces of infinite volume with negative curvature

Abstract

We prove that the imaginary parts of scattering resonances for negatively curved asymptotically hyperbolic surfaces are uniformly bounded away from zero and provide a resolvent bound in the resulting resonance-free strip. This provides an essential spectral gap without the pressure condition. This is done by adapting the methods of [arXiv:1004.3361], [arXiv:1012.4391] and [arXiv:2201.08259] and answers a question posed in [arXiv:1504.06589].

Added an appendix discussing the spectral gap for convex cocompact hyperbolic manifolds in higher dimensions