Invariant measures and lower Ricci curvature bounds

TL;DR

This paper proves the existence of a G-invariant measure equivalent to the original in RCD^*(K,N) spaces under isometry group actions, with applications to Lie groups, homogeneous, and symmetric spaces.

Contribution

It establishes the existence of G-invariant measures preserving RCD^*(K,N) conditions in metric measure spaces with isometry group actions.

Findings

Existence of G-invariant measure equivalent to original measure.

Preservation of RCD^*(K,N) condition under G-invariance.

Dimensional gaps for subgroups of isometries.

Abstract

Given a metric measure space $(X,d,\mathfrak{m})$ that satisfies the Riemannian Curvature Dimension condition, $RCD^*(K,N),$ and a compact subgroup of isometries $G \leq Iso(X)$ we prove that there exists a $G-$invariant measure, $\mathfrak{m}_G,$ equivalent to $\mathfrak{m}$ such that $(X,d,\mathfrak{m}_G)$ is still a $RCD^*(K,N)$ space. We also obtain some applications to Lie group actions on $RCD^*(K,N)$ spaces. We look at homogeneous spaces, symmetric spaces and obtain dimensional gaps for closed subgroups of isometries.