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

TL;DR

This paper characterizes non-collapsed $RCD(K, N)$ spaces by examining the divergence-free property of the second principal term in the heat kernel metric expansion, revealing a new criterion for non-collapsing in these spaces.

Contribution

It establishes a novel characterization of non-collapsed $RCD(K, N)$ spaces through the divergence-free property of a specific metric expansion term, even for weighted Riemannian manifolds.

Findings

Divergence-free property of the second principal term characterizes non-collapsed $RCD(K, N)$ spaces.

The result is sharp and does not extend to noncompact spaces.

Provides new insights into the structure of $RCD(K, N)$ spaces via heat kernel analysis.

Abstract

We investigate the second principal term in the expansion of metrics $c(n)t^{(n+2)/2}g_t$ induced by heat kernel embedding into $L^2$ 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.