A new renormalized volume type invariant

TL;DR

This paper introduces a novel conformal invariant for complete non-compact hyperbolic surfaces, providing a new geometric criterion for selecting canonical representations of bounded domains in the complex plane.

Contribution

It defines a new renormalized volume type invariant and computes it for one-connected surfaces, advancing the understanding of conformal geometry of hyperbolic surfaces.

Findings

New conformal invariant for hyperbolic surfaces

Computed invariant for one-connected surfaces

Provides geometric criteria for canonical domain representations

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 $\mathbb{C}$.