A new renormalized volume type invariant

TL;DR

The paper develops a renormalized volume-type conformal invariant $\Lambda(M,g_M)$ for complete non-compact hyperbolic surfaces that admit a conformal compactification to a bounded domain in $\mathbb{C}$, linking it to Shen–Wang's functional $\lambda(\Omega,v)$ and the boundary expansion data. It proves a sharp inequality for the doubly-connected case, giving an explicit formula $\Lambda(M,g_M)=\frac{2\pi^2}{3}\left[\left(\frac{\pi}{\ln \beta}\right)^2+1\right]$ in terms of the annulus modulus $\beta$, and demonstrates rigidity: equality holds only when the domain is the canonical annulus up to translation and dilation. The approach combines annulus models, Fourier analysis, and detailed boundary expansions to extend Shen–Wang's results to multiply-connected hyperbolic surfaces and to provide a canonical representation criterion for bounded domains in $\mathbb{C}$. An appendix ensures existence results for Liouville problems on punctured domains, completing the analytical underpinning of the construction.

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 $\mathbb{C}$.

Theorems & Definitions (23)

• Theorem 1.1: MR4308060*Theorem 1.1, 1.2, 1.3
• Definition 1.2
• Theorem 1.3
• Proposition 1.4
• Theorem 2.1: Kellogg-Warschawski Theorem
• Lemma 2.2
• proof
• Remark 3.1
• Lemma 3.2
• proof