A characterization of non-collapsed RCD(K, N) spaces via Einstein tensors

2020/10/06 by Honda, Shouhei, Zhu, Xingyu
#Differential Geometry (math.DG) #FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2010.02530

Abstract

We investigate the second principal term in the expansion of metrics c(n)t^(n+2)/2g_t induced by heat kernel embedding into L² on a compact RCD(K, N) space. We prove that the divergence free property of this term in the weak, asymptotic sense if and only if the space is non-collapsed up to multiplying a constant to the reference measure. This seems new even for weighted Riemannian manifolds. Moreover an example tells us that the result cannot be generalized to the noncompact case. In this sense, our result is sharp.

Related

• Non-collapsed spaces with Ricci curvature bounded from below
• Weakly non-collapsed RCD spaces are strongly non-collapsed
• Singular Kähler-Einstein metrics and RCD spaces
• Hypercontractivity of the heat flow on \sf RCD(0,N) spaces: sharpness and rigidities
• Volume bounds for the quantitative singular strata of non collapsed RCD metric measure spaces