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We show that the only metric measure space with the structure of an -cone and with two-sided synthetic Ricci bounds is the Euclidean space for 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 -transport distance. Moreover, we establish rigidity results of independent interest which characterize the -dimensional standard sphere as the unique minimizer of
among all metric measure spaces with dimension bounded above by and Ricci curvature bounded below by .
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Matthias Erbar, Karl-Theodor Sturm, Rigidity of cones with bounded Ricci curvature. J. Eur. Math. Soc. 23 (2021), no. 1, pp. 219–235DOI 10.4171/JEMS/1010