[Paper Review] Fractal Weyl laws for asymptotically hyperbolic manifolds

Kiril Datchev, Semyon Dyatlov|arXiv (Cornell University)|Jun 11, 2012

Spectral Theory in Mathematical Physics33 references4 citations

TL;DR

This paper establishes a fractal Weyl law for resonances on asymptotically hyperbolic manifolds with hyperbolic trapped sets, proving an upper bound on the number of resonances near the essential spectrum proportional to $ (1+|t|)^{eta} $, where $ \beta $ is the Hausdorff dimension of the trapped set. The result generalizes prior work to convex cocompact quotients of hyperbolic space, including quasifuchsian groups, and applies to both constant and nonconstant curvature settings via a novel combination of Vasy's meromorphic continuation and functional calculus techniques.

ABSTRACT

For asymptotically hyperbolic manifolds with hyperbolic trapped sets we prove a fractal upper bound on the number of resonances near the essential spectrum, with power determined by the dimension of the trapped set. This covers the case of general convex cocompact quotients (including the case of connected trapped sets) where our result implies a bound on the number of zeros of the Selberg zeta function in disks of arbitrary size along the imaginary axis. Although no sharp fractal lower bounds are known, the case of quasifuchsian groups, included here, is most likely to provide them.

Motivation & Objective

• To establish a fractal upper bound on the number of resonances near the essential spectrum for asymptotically hyperbolic manifolds with hyperbolic trapped sets.
• To generalize the fractal Weyl law beyond Schottky groups to general convex cocompact quotients of hyperbolic space.
• To extend the result to asymptotically hyperbolic manifolds with non-constant curvature, under the evenness condition on the metric.
• To unify Vasy's effective meromorphic continuation with the resonance counting framework of Sjöstrand–Zworski using functional calculus.
• To provide a sharp upper bound on the number of zeros of the Selberg zeta function in disks along the imaginary axis, with power determined by the limit set's Hausdorff dimension.

Proposed method

• Uses Vasy's method of effective meromorphic continuation of the resolvent to extend the Laplacian resolvent meromorphically to $ \mathbb{C} $, with poles at resonances.
• Applies functional calculus techniques to replace the complex second microlocalization used in Sjöstrand–Zworski, simplifying the counting on $ h $-size scales.
• Constructs Lyapunov/escape functions explicitly in the model case of the hyperbolic cylinder to control dynamics near the trapped set and its incoming/outgoing tails.
• Introduces a logarithmic flattening of the difference $ \varphi_- - \varphi_+ $, where $ \varphi_\pm $ measure distance to the incoming/outgoing tails, to define a global escape function $ \hat{f} $ with uniform lower bounds away from the trapped set.
• Uses the geodesic flow's hyperbolicity on the trapped set to define a Hamiltonian $ p_0 $ and analyze its dynamics via the associated vector field $ H_{p_0} $.
• Implements a cutoff-based construction of the escape function $ f_0 $ to ensure it is smooth and strictly increasing along flowlines outside a neighborhood of the trapped set.

Experimental results

Research questions

• RQ1 What is the precise fractal upper bound on the number of resonances near the essential spectrum for asymptotically hyperbolic manifolds with hyperbolic trapped sets?
• RQ2 How does the exponent in the resonance counting bound relate to the geometric dimension of the trapped set?
• RQ3 Can the fractal Weyl law be extended beyond Schottky groups to general convex cocompact quotients of hyperbolic space?
• RQ4 To what extent can the functional calculus approach replace second microlocalization in resonance counting?
• RQ5 What is the relationship between the Selberg zeta function's zeros and the resonances in the non-constant curvature setting?

Key findings

• The number of resonances in a disk of radius $ R $ centered at $ it $ satisfies the upper bound $ \sum_{|s-it|<R} m_\Gamma(s) \leq C(1+|t|)^{\delta_\Gamma} $, where $ \delta_\Gamma $ is the Hausdorff dimension of the limit set.
• The result holds for all convex cocompact quotients $ \Gamma\backslash\mathbb{H}^n $, including quasifuchsian groups with connected, fractal limit sets and $ \delta_\Gamma > 1 $, extending beyond the Schottky case.
• The bound is valid even when the curvature is not constant, provided the metric is asymptotically hyperbolic and $ g_1 $ is even in $ \tilde{x}^2 $.
• The authors construct explicit escape functions $ \hat{f} $ and $ f_0 $ that are compatible with both Vasy's continuation method and the functional calculus approach.
• The proof simplifies the counting argument by replacing second microlocalization with adapted functional calculus, improving transparency and applicability.
• The framework applies to the hyperbolic cylinder model, where the trapped set and its incoming/outgoing tails are explicitly described, and the escape functions are constructed via direct computation of Hamiltonian flows.