2024/11/07 by Connell, Chris, Dai, Xianzhe, Núñez-Zimbrón, Jesús +3 · 1 citation
#46E36 #53C23 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2411.04327
We develop the barycenter technique of Besson--Courtois--Gallot so that it can be applied on RCD metric measure spaces. Given a continuous map f from a non-collapsed RCD(-(N-1),N) space X without boundary to a locally symmetric N-manifold we show a version of BCG's entropy-volume inequality. The lower bound involves homological and homotopical indices which we introduce. We prove that when equality holds and these indices coincide X is a locally symmetric manifold, and f is homotopic to a Riemannian covering whose degree equals the indices. Moreover, we show a measured Gromov--Hausdorff stability of X and Y involving the homotopical invariant. As a byproduct, we extend a Lipschitz volume rigidity result of Li--Wang to RCD(K,N) spaces without boundary. Finally, we include an application of these methods to the study of Einstein metrics on 4-orbifolds.