Large deviations for positive curvature in Liouville uniformization

Establish a large deviation principle for the Liouville action’s conformal factor on compact surfaces of positive curvature, analogous to the results proved for negatively curved surfaces, thereby extending the large-deviation analysis of fluctuations around metrics of uniform positive curvature.

Background

In the classical regime Q=2/γ, critical points of the Liouville action correspond to metrics conformal to a reference metric with uniform curvature. Large-deviation techniques have been used to analyze this correspondence for negatively curved surfaces.

The authors note that the analogous positive curvature case has not yet been treated, identifying it as an outstanding gap in the understanding of large-deviation behavior for Liouville-type fluctuations.