Accepted Paper

Inserted: 7 oct 2021
Last Updated: 11 oct 2022

Journal: J. Reine Angew. Math. (Crelle's Journal)
Year: 2021

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 $\quad$- $\mathrm{tr}(\mathrm{Hess}f)=\Delta f$ on $U\subset\mathsf{X}$ for every $f$ sufficiently regular, $\quad$- $\mathfrak{m}=c\mathscr{H}^n$ 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 $n$.