On a singular Liouville-type equation and the Alexandrov isoperimetric inequality

TL;DR

This paper extends an inequality related to singular Liouville equations, providing new proofs and characterizations for the Alexandrov isoperimetric inequality on singular surfaces, with implications for metrics of bounded curvature.

Contribution

It offers a generalized inequality for weak subsolutions of singular Liouville equations and introduces new characterizations of local metrics on Alexandrov surfaces.

Findings

New proof of the Alexandrov isoperimetric inequality on singular surfaces

Characterization of local metrics on Alexandrov's surfaces of bounded curvature

Novel insights into the equality case in the isoperimetric inequality

Abstract

We obtain a generalized version of an inequality, first derived by C. Bandle in the analytic setting, for weak subsolutions of a singular Liouville-type equation. As an application we obtain a new proof of the Alexandrov isoperimetric inequality on singular abstract surfaces. Interestingly enough, motivated by this geometric problem, we obtain a seemingly new characterization of local metrics on Alexandrov's surfaces of bounded curvature. At least to our knowledge, the characterization of the equality case in the isoperimetric inequality in such a weak framework is new as well.