Curvature-Dimension for Autonomous Lagrangians

TL;DR

This paper develops a curvature-dimension condition for autonomous Lagrangians on weighted manifolds, linking it to displacement convexity of entropy and extending classical geometric inequalities to new settings.

Contribution

It introduces a novel curvature-dimension condition for Lagrangians, generalizes Klartag's needle decomposition, and applies these ideas to derive new geometric inequalities.

Findings

Curvature-dimension condition is equivalent to displacement convexity of entropy.

Generalization of Klartag's needle decomposition to Lagrangian setting.

Extension of Brunn-Minkowski inequalities to complex hyperbolic space and contact magnetic geodesics.

Abstract

We introduce a curvature-dimension condition for autonomous Lagrangians on weighted manifolds, which depends on the Euler-Lagrange dynamics on a single energy level. By generalizing Klartag's needle decomposition technique to the Lagrangian setting, we prove that this curvature-dimension condition is equivalent to displacement convexity of entropy along cost-minimizing interpolations in an $L^1$ sense, and that it implies various consequences of lower Ricci curvature bounds, as in the metric setting. As examples we consider classical and isotropic Lagrangians on Riemannian manifolds. In particular, we generalize the horocyclic Brunn-Minkowski inequality to complex hyperbolic space of arbitrary dimension, and present a new Brunn-Minkowski inequality for contact magnetic geodesics on odd-dimensional spheres.