Displacement convexity of Invariant Measures and curvature bounds of isometric actions

Abstract

Let $G\curvearrowright M$ be a proper isometric action of a compact Lie group on a complete, connected and orientable Riemannian manifold of dimension $N$. We characterize the local $K$-displacement convexity of the internal-energy functional $H$ on the space $\mathcal P^{ac}_G(M)$ of absolutely continuous $G$-invariant probability measures. Via disintegration along the principal orbits, $H$ reduces to the internal energy of a transversal density against the orbit-volume--weighted measure $\mathfrak m = V\operatorname{vol}_{M/G}$, and the convexity of $H$ is equivalent to the $N$-Bakry--\'Emery condition $\operatorname{Ric}^{\Psi}_N \ge K$ on the weighted quotient $(M/G,\mathfrak m)$. Written on $M$, this bound reads $\operatorname{Ric}^{\mathcal H}_M(v) + 3\|A_v\|^2 - \operatorname{Hess}(\log V)(v,v) - \langle\vec H,v\rangle^2/m \ge K|v|^2$ at every principal point and horizontal direction $v$, where $A$ is the O'Neill integrability tensor of $\pi\colon M\to M/G$, $\vec H$ the mean-curvature vector of the orbits, and $m=\dim(G\cdot x)$ the orbit dimension. The terms on the left encode, in this order, the horizontal curvature of $M$, the non-integrability of the horizontal distribution, and --- in the last two --- the variation of the orbit volume. When the orbits are points one recovers the theorem of von Renesse--Sturm.