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.
We extend the notion of Plancherel convergence, introduced in the study of compact quotients of locally compact groups, to finite-area hyperbolic surfaces and establish a quantum ergodicity theorem for Plancherel sequences. We show that the expectation values of integral operators whose kernels are uniformly bounded an...
Consider a holomorphic family (fλ)λ∈Λ$(f_\lambda)_{\lambda \in \Lambda }$ of polynomial maps on C$\mathbb {C}$ with the property that a critical point of fλ$f_\lambda$ is persistently preperiodic to a repelling periodic point of fλ$f_\lambda$ . Let Ω$\Omega$ be a bounded stable component of Λ$\Lambda$ with the property...
F. Bianchi, Y. He· Journal of the London Mathem...· 2 citations
In this paper, we extend a large part of the uniformization theory of Bonk-Heinonen-Koskela [Asterisque 2001] to length spaces that are not necessarily proper or geodesic. Among other things, we show that there is a one-to-one correspondence between the quasiisometry classes of complete roughly starlike Gromov hyperbol...
We show that the sharp Euclidean isoperimetric inequality holds for domains in complete simply connected Riemannian $5$-manifolds of nonpositive sectional curvature, which establishes the Cartan-Hadamard conjecture in that dimension. The main step is a sharp inequality for constant-mean-curvature hypersurfaces, proved...
Shi-Bing Chen, M. Ghomi, Peng Wang· 3 citations· ⚡1
We study spectral zeta densities associated with complex powers of principally scalar Laplace-type operators on pseudo-Riemannian scattering spaces of mixed signature and arbitrary dimension. The construction extends, beyond the even-dimensional Lorentzian case, the local analysis of Feynman complex powers developed by...
We describe a version of positive macroscopic scalar curvature motivated by the work of Alpert, Balitskiy, and Guth, and prove that this condition on a manifold implies a bound on its 1-width in terms of its first Betti number. A key tool in the proof is a decomposition of any closed manifold into a family of chains wi...
Brendan Isley· 0 citations
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