Large deviations for Liouville theory on positively curved surfaces

Establish large deviation principles for the Liouville path integral associated with the action S_g(φ) on compact Riemann surfaces of positive curvature, analogous to the results proved for negatively curved surfaces, thereby characterizing fluctuations around constant positive curvature metrics.

Background

In the classical regime Q=2/γ, critical points of the Liouville action correspond to metrics of constant curvature. Large deviation techniques have been used to paper fluctuations around such metrics on negatively curved surfaces. However, the corresponding analysis for positively curved surfaces has not been achieved.

Clarifying the large deviations in positive curvature would complete the picture of metric fluctuations encoded by the Liouville action across all curvature regimes.