An improved geometric inequality via vanishing moments, with applications to singular Liouville equations

TL;DR

This paper improves the geometric inequality related to singular Liouville equations by analyzing angular distributions near singularities, leading to new existence and non-existence results in geometric analysis and mathematical physics.

Contribution

It introduces an enhanced inequality via vanishing moments that refines the Moser-Trudinger inequality for singular Liouville equations, with applications to various physical models.

Findings

Improved inequality bounds for singular Liouville equations.

Conditions for existence and non-existence of solutions.

Applications to models in physics like Chern-Simons theories.

Abstract

We consider a class of singular Liouville equations on compact surfaces motivated by the study of Electroweak and Self-Dual Chern-Simons theories, the Gaussian curvature prescription with conical singularities and Onsager's description of turbulence. We analyse the problem of existence variationally, and show how the angular distribution of the conformal volume near the singularities may lead to improvements in the Moser-Trudinger inequality, and in turn to lower bounds on the Euler-Lagrange functional. We then discuss existence and non-existence results.