  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.

Differential Geometry;Analysis of PDEs