Bounds for Green's functions on noncompact hyperbolic Riemann orbisurfaces of finite volume

TL;DR

This paper extends bounds for Green's functions from compact to noncompact hyperbolic Riemann orbisurfaces of finite volume, providing optimal estimates for these functions in more general geometric settings.

Contribution

It generalizes existing bounds for Green's functions to noncompact hyperbolic Riemann orbisurfaces of finite volume, using methods from prior work on compact surfaces.

Findings

Derived optimal bounds for Green's functions on noncompact hyperbolic Riemann orbisurfaces.

Extended previous bounds from compact to noncompact cases.

Applicable to surfaces of genus greater than zero.

Abstract

In 2006, in a paper published in Compositio titled "Bounds on canonical Green's functions", J. Jorgenson and J. Kramer derived bounds for the canonical Green's function and the hyperbolic Green's function defined on a compact hyperbolic Riemann surface. In this article, we extend these bounds to noncompact hyperbolic Riemann orbisurfaces of finite volume and of genus greater than zero, which can be realized as a quotient space of the action of a Fuchsian subgroup of first kind on the hyperbolic upper half-plane. Our bounds are optimally derived by following the methods from the above mentioned paper of J. Jorgenson and J. Kramer.