Curvature-Dimension for Autonomous Lagrangians

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.