Convex Co-compact Hyperbolic Manifolds Research Articles

Exponential mixing of frame flows for convex cocompact hyperbolic manifolds

The aim of this paper is to establish exponential mixing of frame flows for convex cocompact hyperbolic manifolds of arbitrary dimension with respect to the Bowen–Margulis–Sullivan measure. Some immediate applications include an asymptotic formula for matrix coefficients with an exponential error term as well as the exponential equidistribution of holonomy of closed geodesics. The main technical result is a spectral bound on transfer operators twisted by holonomy, which we obtain by building on Dolgopyat's method.

Volumes of quasifuchsian manifolds

Quasifuchsian hyperbolic manifolds, or more generally convex co-compact hyperbolic manifolds, have infinite volume, but they have a well-defined ``renormalized'' volume. We outline some relations between this renormalized volume and the volume, or more precisely the ``dual volume'', of the convex core. On one hand, there are striking similarities between them, for instance in their variational formulas. On the other, object related to them tend to be within bounded distance. Those analogies and proximities lead to several questions. Both the renormalized volume and the dual volume can be used for instance to bound the volume of the convex core in terms of the Weil-Petersson distance between the conformal metrics at infinity.

Tomas–Stein Restriction Estimates on Convex Cocompact Hyperbolic Manifolds

Abstract In this paper, we investigate the Tomas–Stein restriction estimates on convex cocompact hyperbolic manifolds $\Gamma \backslash{\mathbb{H}}^{n+1}$. Via the spectral measure of the Laplacian, we prove that the Tomas–Stein restriction estimate holds when the limit set has Hausdorff dimension $\delta _\Gamma <n/2$. This provides an example for which restriction estimate holds in the presence of hyperbolic geodesic trapping.

Improved fractal Weyl bounds for hyperbolic manifolds (with an appendix by David Borthwick, Semyon Dyatlov and Tobias Weich)

We give a new fractal Weyl upper bound for resonances of convex co-compact hyperbolic manifolds in terms of the dimension $n$ of the manifold and the dimension $\delta$ of its limit set. More precisely, we show that as $R\to\infty$, the number of resonances in the box $[R,R+1]+i[-\beta,0]$ is $O(R^{m(\beta,\delta)+})$, where the exponent $m(\beta,\delta)=\min(2\delta+2\beta+1-n,\delta)$ changes its behavior at $\beta=(n-1-\delta)/2$. In the case $\delta<(n-1)/2$, we also give an improved resolvent upper bound in the standard resonance free strip $\{\mathrm{Im}\ \lambda\ > \delta-(n-1)/2\}$. Both results use the fractal uncertainty principle point of view recently introduced in [arXiv:1504.06589]. The appendix presents numerical evidence for the Weyl upper bound.

Ruelle and Quantum Resonances for Open Hyperbolic Manifolds

Abstract We establish a direct classical-quantum correspondence on convex cocompact hyperbolic manifolds between the spectra of the geodesic flow and the Laplacian acting on natural tensor bundles. This extends previous work detailing the correspondence for cocompact quotients.

Spectral gaps, additive energy, and a fractal uncertainty principle

We obtain an essential spectral gap for $n$-dimensional convex co-compact hyperbolic manifolds with the dimension $\delta$ of the limit set close to $(n-1)/2$. The size of the gap is expressed using the additive energy of stereographic projections of the limit set. This additive energy can in turn be estimated in terms of the constants in Ahlfors-David regularity of the limit set. Our proofs use new microlocal methods, in particular a notion of a fractal uncertainty principle.

Moments of Eisenstein Series on Convex Co-Compact Hyperbolic Manifolds

Moments of Eisenstein Series on Convex Co-Compact Hyperbolic Manifolds

Equidistribution of Eisenstein series on convex co-compact hyperbolic manifolds

For convex co-compact hyperbolic manifolds $\Gamma\backslash{\Bbb H}^{n+1}$ for which the dimension of the limit set satisfies $\delta_\Gamma&lt; n/2$, we show that the high-frequency Eisenstein series associated to a point $\xi$ ``at infinity'' concentrate microlocally on a measure supported by (the closure of) the set of points in the unit cotangent bundle corresponding to geodesics ending at $\xi$. The average in $\xi$ of these limit measures is the Liouville measure.

Eta invariant and Selberg zeta function of odd type over convex co-compact hyperbolic manifolds

We show meromorphic extension and give a complete description of the divisors of a Selberg zeta function of odd type Z Γ , Σ o ( λ ) associated to the spinor bundle Σ on an odd dimensional convex co-compact hyperbolic manifold Γ \\ H 2 n + 1 . As a byproduct we do a full analysis of the spectral and scattering theory of the Dirac operator on asymptotically hyperbolic manifolds. We show that there is a natural eta invariant η ( D ) associated to the Dirac operator D over a convex co-compact hyperbolic manifold Γ \\ H 2 n + 1 and that exp ( π i η ( D ) ) = Z Γ , Σ o ( 0 ) , thus extending Millson's formula to this setting. Under some assumption on the exponent of convergence of Poincaré series for the group Γ, we also define an eta invariant for the odd signature operator, and we show that for Schottky 3-dimensional hyperbolic manifolds it gives the argument of a holomorphic function which appears in the Zograf factorization formula relating two natural Kähler potentials for Weil–Petersson metric on Schottky space.

Dynamics of asymptotically hyperbolic manifolds

We prove a dynamical wave trace formula for asymptotically hyperbolic (n + 1)-dimensional manifolds with negative (but not necessarily constant) sectional curvatures; the formula equates the renormalized wave trace to the lengths of closed geodesics. This result generalizes the classical theorem of Duistermaat and Guillemin for compact manifolds and the results of Guillope and Zworski, Perry, and Guillarmou and Naud for hyperbolic manifolds with infinite volume. A corollary of this dynamical trace formula is a dynamical resonance-wave trace formula for compact perturbations of convex cocompact hyperbolic manifolds. We define a dynamical zeta function and prove its analyticity in a half plane. In our main result, we produce a prime orbit theorem for the geodesic flow. This is the first such result for manifolds that have neither constant curvature nor finite volume. As a corollary to the prime orbit theorem, using our dynamical resonance-wave trace formula, we show that the existence of pure point spectrum for the Laplacian on negatively curved compact perturbations of convex cocompact hyperbolic manifolds is related to the dynamics of the geodesic flow.

Wave Decay on Convex Co-Compact Hyperbolic Manifolds

For convex co-compact hyperbolic quotients $${X=\Gamma\backslash\mathbb {H}^{n+1}}$$ , we analyze the long-time asymptotic of the solution of the wave equation u(t) with smooth compactly supported initial data f = (f 0, f 1). We show that, if the Hausdorff dimension δ of the limit set is less than n/2, then $${u(t)=C_\delta(f)e^{(\delta-\frac{n}{2})t}/\Gamma(\delta-n/2+1)+e^{(\delta-\frac{n}{2})t}R(t),}$$ where $${C_{\delta}(f)\in C^\infty(X)}$$ and $${||R(t)||=\mathcal{O}(t^{-\infty})}$$ . We explain, in terms of conformal theory of the conformal infinity of X, the special cases $${\delta\in n/2-\mathbb {N}}$$ , where the leading asymptotic term vanishes. In a second part, we show for all $${\epsilon > 0}$$ the existence of an infinite number of resonances (and thus zeros of Selberg zeta function) in the strip $${\{-n\delta-\epsilon < \rm Re(\lambda) < \delta\}}$$ . As a byproduct we obtain a lower bound on the remainder R(t) for generic initial data f.

Wave 0-trace and Length Spectrum on Convex Co-compact Hyperbolic Manifolds

For quotients of the $n+1$-dimensional hyperbolic space by a convex co-compact group $\Gamma$, we obtain a formula relating the renormalized trace of the wave operator with the resonances of the Laplacian and some conformal invariants of the boundary, generalizing a formula of Guillope and Zworski in dimension 2. By writing this trace with the length spectrum, we prove precise asymptotics of the number of closed geodesics with an effective, exponentially small error term when the dimension of the limit set of $\Gamma$ is greater than $n/2$.

A poisson summation formula and lower bounds for resonances in hyperbolic manifolds

For convex co-compact hyperbolic manifolds of even dimension n + 1, we derive a Poisson-type formula for scattering resonances which may be regarded as a version of Selberg's trace formula for these manifolds. Using techniques of Guillop e and Zworski we easily obtain anO R n+1 lower bound for the counting function for scattering resonances together with other lower bounds for the counting function of resonances in strips.

Closed Geodesics in Homology Classes for Convex Co-Compact Hyperbolic Manifolds

Let Γ be a convex co-compact, torsion-free, discrete group of isometries of real hyperbolic space Hn+1. We compute the asymptotics of the counting function for closed geodesics in homology classes for the quotient manifold X = Γ\Hn+1, under the assumption that H1(X, Z) is infinite. Our results imply asymptotic equipartition of geodesics in distinct homology classes.