Calculus of Variations and Geometric Measure Theory
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C. Brena - N. Gigli - S. Honda - X. Zhu

Weakly non-collapsed RCD spaces are strongly non-collapsed

created by brena on 07 Oct 2021



Inserted: 7 oct 2021

Year: 2021

ArXiv: 2110.02420 PDF


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 $n$.

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