## Analysis & PDE

• Anal. PDE
• Volume 5, Number 3 (2012), 513-552.

### Sharp geometric upper bounds on resonances for surfaces with hyperbolic ends

David Borthwick

#### Abstract

We establish a sharp geometric constant for the upper bound on the resonance counting function for surfaces with hyperbolic ends. An arbitrary metric is allowed within some compact core, and the ends may be of hyperbolic planar, funnel, or cusp type. The constant in the upper bound depends only on the volume of the core and the length parameters associated to the funnel or hyperbolic planar ends. Our estimate is sharp in that it reproduces the exact asymptotic constant in the case of finite-area surfaces with hyperbolic cusp ends, and also in the case of funnel ends with Dirichlet boundary conditions.

#### Article information

Source
Anal. PDE, Volume 5, Number 3 (2012), 513-552.

Dates
Received: 31 July 2010
Accepted: 26 February 2011
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.apde/1513731229

Digital Object Identifier
doi:10.2140/apde.2012.5.513

Mathematical Reviews number (MathSciNet)
MR2994506

Zentralblatt MATH identifier
1264.35157

#### Citation

Borthwick, David. Sharp geometric upper bounds on resonances for surfaces with hyperbolic ends. Anal. PDE 5 (2012), no. 3, 513--552. doi:10.2140/apde.2012.5.513. https://projecteuclid.org/euclid.apde/1513731229

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