Microlocal analysis and spectral gap for complex hyperbolic manifolds

Abstract

We prove the existence of an essential spectral gap for convex cocompact complex hyperbolic manifolds under the hypothesis that the limit set does not contain a complex circle. We apply an extension to complex hyperbolic manifolds due to Quan of the approach of Vasy to meromorphic continuation of the resolvent. This allows us to adapt the method of Dyatlov-Zahl to prove fine microlocal properties of resonant states. Following recent work of Athreya-Dyatlov-Miller and Dyatlov-Jezequel, we then take advantage of the different expansion rates in different directions for the geodesic flow, and apply the one dimensional fractal uncertainty principle of Bourgain-Dyatlov along the fast expanding direction. Combining these techniques provides a new resolvent bound from which the essential spectral gap follows.