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Spectral gap for surfaces of infinite volume with negative curvature

Zhongkai Tao

TL;DR

This work proves a uniform spectral gap for scattering resonances on negatively curved, infinite-volume, asymptotically hyperbolic surfaces by extending the Bourgain–Dyatlov paradigm without a pressure condition. The authors develop a weight-free Grushin framework to construct quantum monodromy maps via microlocal and global Grushin problems and exploit Vasy’s meromorphic continuation, together with a one-dimensional trapped set arising from negative curvature. The key technical advance is relating resonances to zeros of a finite-dimensional determinant, then leveraging a fractal uncertainty principle to bound the monodromy operator powers, yielding both a resonance-free strip and a resolvent bound. The results generalize spectral-gap phenomena in open hyperbolic systems to a broader geometric setting and provide a sharp, conditional-free gap in dimension two with implications for spectral theory and scattering on asymptotically hyperbolic manifolds.

Abstract

We prove that the imaginary parts of scattering resonances for negatively curved asymptotically hyperbolic surfaces are uniformly bounded away from zero and provide a resolvent bound in the resulting resonance-free strip. This provides an essential spectral gap without the pressure condition. This is done by adapting the methods of [arXiv:1004.3361], [arXiv:1012.4391] and [arXiv:2201.08259] and answers a question posed in [arXiv:1504.06589].

Spectral gap for surfaces of infinite volume with negative curvature

TL;DR

This work proves a uniform spectral gap for scattering resonances on negatively curved, infinite-volume, asymptotically hyperbolic surfaces by extending the Bourgain–Dyatlov paradigm without a pressure condition. The authors develop a weight-free Grushin framework to construct quantum monodromy maps via microlocal and global Grushin problems and exploit Vasy’s meromorphic continuation, together with a one-dimensional trapped set arising from negative curvature. The key technical advance is relating resonances to zeros of a finite-dimensional determinant, then leveraging a fractal uncertainty principle to bound the monodromy operator powers, yielding both a resonance-free strip and a resolvent bound. The results generalize spectral-gap phenomena in open hyperbolic systems to a broader geometric setting and provide a sharp, conditional-free gap in dimension two with implications for spectral theory and scattering on asymptotically hyperbolic manifolds.

Abstract

We prove that the imaginary parts of scattering resonances for negatively curved asymptotically hyperbolic surfaces are uniformly bounded away from zero and provide a resolvent bound in the resulting resonance-free strip. This provides an essential spectral gap without the pressure condition. This is done by adapting the methods of [arXiv:1004.3361], [arXiv:1012.4391] and [arXiv:2201.08259] and answers a question posed in [arXiv:1504.06589].
Paper Structure (14 sections, 19 theorems, 86 equations)

This paper contains 14 sections, 19 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Microlocal preliminaries
  3. Review of concepts
  4. Vasy's method revisited
  5. Quantum monodromy without weights
  6. Construction of quantum monodromy maps
  7. Dynamical preliminaries
  8. Microlocal Grushin problem
  9. Global Grushin problem
  10. Structure of the trapped set
  11. Proof of the spectral gap and the resolvent estimate
  12. Spectral gap for convex cocompact hyperbolic manifolds
  13. Convex cocompact hyperbolic manifolds
  14. Spectral gap

Key Result

Theorem 1

Suppose $X$ has dimension $2$ and (strictly) negative curvature. Then there exist $C_0,\beta>0$ such that the resolvent continues holomorphically from $\mathop{\rm Im}\nolimits \lambda> 1$ to $\{|\lambda|>C_0,\, \mathop{\rm Im}\nolimits \lambda>-\beta\}$. Moreover, for any $\chi\in C_c^\infty(X)$, we have the resolvent bound for $\mathop{\rm Im}\nolimits \lambda>-\beta$ and $|\lambda|>C_0$.

Theorems & Definitions (29)

  • Theorem
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Lemma 2.4
  • proof
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • ...and 19 more