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

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