Magnetic Brunn-Minkowski and Borell-Brascamp-Lieb inequalities on Riemannian manifolds

Abstract

We study the Brunn-Minkowski and Borell-Brascamp-Lieb inequalities on a Riemannian manifold endowed with an exact magnetic field, replacing geodesics by minimizers of an action functional given by the length minus the integral of the magnetic potential. We prove that nonnegativity of a suitably defined magnetic Ricci curvature implies a magnetic Brunn-Minkowski inequality. More generally, given an arbitrary volume form on the manifold, we introduce a weighted magnetic Ricci curvature, and prove a magnetic version of the Borell-Brascamp-Lieb inequality.