Rigidity of cones with bounded Ricci curvature

TL;DR

This paper proves that the only N-cone metric measure space with two-sided synthetic Ricci bounds is Euclidean space, and characterizes the sphere as a unique minimizer of a cosine integral among spaces with bounded dimension and Ricci curvature.

Contribution

It introduces a new notion of Ricci curvature upper bounds via heat kernel asymptotics and establishes rigidity results characterizing Euclidean space and spheres.

Findings

Euclidean space is the only N-cone with two-sided synthetic Ricci bounds.

The N-sphere uniquely minimizes a cosine integral among spaces with bounded dimension and Ricci curvature.

A novel notion of Ricci curvature upper bounds based on heat kernel asymptotics is proposed.

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 asymtotic of the heat kernel in the $L^2$-transport distance. Moreover, we establish a beautiful rigidity results of independent interest which characterize the $N$-dimensional standard sphere ${\mathbb S}^N$ as the unique minimizer of $$\int_X\int_X \cos d (x,y)\, m(d y)\,m(d x)$$ among all metric measure spaces with dimension bounded above by $N$ and Ricci curvature bounded below by $N-1$.