Rigidity of cones with bounded Ricci curvature

  • Matthias Erbar

    Universität Bielefeld, Germany
  • Karl-Theodor Sturm

    Universität Bonn, Germany
Rigidity of cones with bounded Ricci curvature cover
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We show that the only metric measure space with the structure of an NN-cone and with two-sided synthetic Ricci bounds is the Euclidean space RN+1\mathbb R^{N+1} for NN 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 L2L^2-transport distance. Moreover, we establish rigidity results of independent interest which characterize the NN-dimensional standard sphere SN\mathbb S^N as the unique minimizer of

XXcosd(x,y)m(dy)m(dx)\int_X\int_X \cos d (x,y)\, m(\mathrm d y)\,m(\mathrm d x)

among all metric measure spaces with dimension bounded above by NN and Ricci curvature bounded below by N1N-1.

Cite this article

Matthias Erbar, Karl-Theodor Sturm, Rigidity of cones with bounded Ricci curvature. J. Eur. Math. Soc. 23 (2021), no. 1, pp. 219–235

DOI 10.4171/JEMS/1010