A new renormalized volume type invariant

Abstract

In this paper, we define a new conformal invariant on complete non-compact hyperbolic surfaces that can be conformally compactified to bounded domains in $\mathbb{C}$. We study and compute this invariant up to one-connected surfaces. Our results give a new geometric criterion for choosing canonical representations of bounded domains in $\mathbb{C}$.

Accepted by Pacific Journal of Mathematics