Poincar\'e series for surfaces with boundary

TL;DR

This paper proves that the Poincaré series for counting orthogeodesics and geodesic arcs on negatively curved surfaces with boundary extend meromorphically across the complex plane and provides their values at zero.

Contribution

It establishes the meromorphic extension of Poincaré series for surfaces with boundary, a novel result in geometric analysis.

Findings

Poincaré series extend meromorphically to the entire complex plane.

Explicit values of the series at zero are obtained.

Results apply to negatively curved surfaces with geodesic boundary.

Abstract

We show that the Poincar\'e 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.