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

@article{Honda2020ACO, title={A characterization of non-collapsed \$RCD(K, N)\$ spaces via Einstein tensors}, author={Shouhei Honda and Xingyu Zhu}, journal={arXiv: Differential Geometry}, year={2020} }

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.

#### 2 Citations

Weakly non-collapsed RCD spaces are strongly non-collapsed

- Mathematics
- 2021

We prove that any weakly non-collapsed RCD space is actually non-collapsed, up to a renormalization of the measure. This confirms a conjecture raised by De Philippis and the second named author in… Expand

Heat flow on 1-forms under lower Ricci bounds. Functional inequalities, spectral theory, and heat kernel.

- Mathematics
- 2020

We study the canonical heat flow $(\mathsf{H}_t)_{t\geq 0}$ on the cotangent module $L^2(T^*M)$ over an $\mathrm{RCD}(K,\infty)$ space $(M,\mathsf{d},\mathfrak{m})$, $K\in\mathbb{R}$. We show… Expand

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