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

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.