Weakly non-collapsed RCD spaces are strongly non-collapsed

Abstract

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 full generality. One of the auxiliary results of independent interest that we obtain is about the link between the properties - $\mathrm{tr}(\mathrm{Hess}f)=\Delta f$ on $U\subset \mathsf{X}$ for every $f$ sufficiently regular, - $\mathfrak{m}=c{\mathcal{H}}^n$ on $U\subset \mathsf{X}$ for some $c>0$, where $U\subset \mathsf{X}$ is open and $\mathsf{X}$ is a - possibly collapsed - RCD space of essential dimension $n$.

arXiv comment31 pages

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Cited by in corpus (2)

• Singular Weyl's law with Ricci curvature bounded below
• On the asymptotic behaviour of the fractional Sobolev seminorms in metric measure spaces: Bourgain-Brezis-Mironescu's theorem revisited