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 ${\mathcal{P}}_G^{ac}(M)$ 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 $\mathfrak{m}=V{\mathrm{vol}}_{M/G}$, and the convexity of $H$ is equivalent to the $N$-Bakry--Émery condition ${\mathrm{Ric}}_N^{\Psi }\ge K$ on the weighted quotient $(M/G,\mathfrak{m})$. Written on $M$, this bound reads ${\mathrm{Ric}}_M^{\mathcal{H}}(v)+3\Vert A_v{\Vert }^2-\mathrm{Hess}(\log V)(v,v)-⟨\vec{H},v{⟩}^2/m\ge K|v{|}^2$ at every principal point and horizontal direction $v$, where $A$ is the O'Neill integrability tensor of $\pi :M\to M/G$, $\vec{H}$ 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.

Revised and extended version, 47 pages