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Rigidity of cones with bounded Ricci curvature
Journal of the European Mathematical Society ( IF 2.6 ) Pub Date : 2020-10-09 , DOI: 10.4171/jems/1010
Matthias Erbar 1 , Karl-Theodor Sturm 2
Affiliation  

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 $N-1$.

中文翻译:

有界 Ricci 曲率锥体的刚性

我们表明,具有 $N$-cone 结构和两侧合成 Ricci 界的唯一度量空间是 $N$ 整数的欧几里得空间 ${\mathbb R}^{N+1}$。这是基于 Ricci 曲率上限的新概念,根据 $L^2$-传输距离中热核的短时间渐近线给出的度量度量空间。此外,我们建立了一个独立兴趣的美丽刚性结果,它将 $N$ 维标准球面 ${\mathbb S}^N$ 表征为 $$\int_X\int_X \cos d (x,y)\ , m(dy)\,m(dx)$$ 在所有度量空间中,维度以 $N$ 为界,Ricci 曲率以 $N-1$ 为界。
更新日期:2020-10-09
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