Liouville's equation for curvature and systolic defect

We analyze the probabilistic variance of a solution of Liouville's equation for curvature, given suitable bounds on the Gaussian curvature. The related systolic geometry was recently studied by Horowitz, Katz, and Katz, where we obtained a strengthening of Loewner's torus inequality containing a "defect term", similar to Bonnesen's strengthening of the isoperimetric inequality. Here the analogous isosystolic defect term depends on the metric and "measures" its deviation from being flat. Namely, the defect is the variance of the function f which appears as the conformal factor expressing the metric on the torus as f^2(x,y)(dx^2+dy^2), in terms of the flat unit-area metric in its conformal class. A key tool turns out to be the computational formula for probabilistic variance, which is a kind of a sharpened version of the Cauchy-Schwartz inequality.