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Poincaré series for surfaces with boundary

Yann Chaubet

Abstract

We show that the Poincaré series counting orthogeodesics of a negatively curved surface with totally geodesic boundary extends meromorphically to the whole complex plane, as well as the series counting geodesic arcs linking two points; we also give their value at zero.

Poincaré series for surfaces with boundary

Abstract

We show that the Poincaré series counting orthogeodesics of a negatively curved surface with totally geodesic boundary extends meromorphically to the whole complex plane, as well as the series counting geodesic arcs linking two points; we also give their value at zero.

Paper Structure

This paper contains 14 sections, 13 theorems, 133 equations.

Table of Contents

  1. Introduction
  2. Geometrical and dynamical preliminaries
  3. Extension to a surface with strictly convex boundary
  4. Structural forms
  5. Extension of the geodesic vector field
  6. Hyperbolicity of the geodesic flow
  7. The resolvent
  8. Poincaré series
  9. Counting measure
  10. A pairing formula for the Poincaré series
  11. Value of the Poincaré series at the origin
  12. Behavior of $Q(s)$ at $s=0$
  13. Value at $s=0$
  14. Poincaré series for geodesic arcs linking two points

Key Result

Theorem 1

The Poincaré series $s \mapsto \eta(s)$ extends meromorphically to the whole complex plane and vanishes at $s=0$.

Theorems & Definitions (27)

  • Theorem 1
  • Theorem 2
  • Lemma 2.1
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Remark 3.3
  • ...and 17 more