# Rigidity of cones with bounded Ricci curvature

### Matthias Erbar

Universität Bielefeld, Germany### Karl-Theodor Sturm

Universität Bonn, Germany

## Abstract

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 $\mathbb R^{N+1}$ 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 $\mathbb S^N$ as the unique minimizer of

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

## 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