Rigidity Of Cones With Bounded Ricci Curvature

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 L-2-transport distance. Moreover, we establish rigidity results of independent interest which characterize the N-dimensional standard sphere S-N as the unique minimizer e.g. ofintegral(X)integral(X) cos(d(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.