Rigidity of cones with bounded Ricci curvature

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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 asymtotic of the heat kernel in the L²-transport distance. Moreover, we establish a beautiful rigidity results of independent interest which characterize the N-dimensional standard sphere Sⁿ as the unique minimizer of ∫ₓ∫ₓ d (x,y) m(d y)m(d x) among all metric measure spaces with dimension bounded above by N and Ricci curvature bounded below by N-1.