Liouville's equation for curvature and systolic defect

TL;DR

This paper explores the variance of solutions to Liouville's equation for curvature and introduces a defect term in systolic geometry that quantifies deviation from flatness using probabilistic variance.

Contribution

It establishes a new isosystolic inequality with a defect term based on the variance of the conformal factor, linking probabilistic methods to geometric analysis.

Findings

Variance of the conformal factor measures deviation from flatness.

Derived a strengthened systolic inequality with a defect term.

Applied probabilistic variance formula to geometric curvature analysis.

Abstract

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.