PUB - Publikationen an der Universität Bielefeld

We show that the only metric measure space with the structure of an N-cone and with two-sided synthetic Ricci bounds is the Euclidean space RN+1 for N integer. This is based on a novel notion of Ricci curvature upper bounds for metric measure spaces given in terms of the short time asymptotic of the heat kernel in the L2-transport distance. Moreover, we establish rigidity results of independent interest which characterize the N-dimensional standard sphere SN as the unique minimizer of ∫X​∫X​cosd(x,y)m(dy)m(dx) among all metric measure spaces with dimension bounded above by N and Ricci curvature bounded below by N−1.