The Rigidity and Gap Theorem for Liouville's Equation

TL;DR

This paper investigates the properties of solutions to Liouville's equation, establishing rigidity and gap theorems that relate boundary integrals to domain geometry, especially the number of boundary components.

Contribution

It introduces new rigidity and gap theorems for boundary integrals in Liouville's equation, linking conformal structure to domain shape and boundary complexity.

Findings

Boundary integral is always nonpositive.

Zero boundary integral characterizes the disc.

Gap theorems relate boundary integral to boundary components.

Abstract

In this paper, we study the properties of the first global term in the polyhomogeneous expansions for Liouville's equation. We obtain rigidity and gap results for the boundary integral of the global coefficient. We prove that such a boundary integral is always nonpositive, and is zero if and only if the underlying domain is a disc. More generally, we prove some gap theorems relating such a boundary integral to the number of components of the boundary. The conformal structure plays an essential role.