AI 中文总结
本文证明凸余紧复双曲流形在极限集不含复圆时存在本质谱隙，通过Quan推广的Vasy方法、Dyatlov-Zahl微局域技术及Bourgain-Dyatlov分形不确定性原理获得新的预解式界。
AI 中文摘要
我们证明了在极限集不包含复圆的假设下，凸余紧复双曲流形存在本质谱隙。我们应用了Quan将Vasy方法推广到复双曲流形以进行预解式的亚纯延拓。这使我们能够调整Dyatlov-Zahl的方法来证明共振态的精细微局域性质。遵循Athreya-Dyatlov-Miller和Dyatlov-Jezequel最近的工作，我们利用测地流在不同方向上不同的扩张速率，并沿快速扩张方向应用Bourgain-Dyatlov的一维分形不确定性原理。结合这些技术提供了新的预解式界，由此得出本质谱隙。
英文摘要
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.
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