Weakly non-collapsed RCD spaces are strongly non-collapsed

TL;DR

This paper proves that weakly non-collapsed RCD spaces are actually non-collapsed after measure renormalization, confirming a conjecture and establishing a link between geometric and measure-theoretic properties.

Contribution

It establishes that weakly non-collapsed RCD spaces are non-collapsed up to measure renormalization, confirming a conjecture and connecting differential properties with measure structure.

Findings

Weakly non-collapsed RCD spaces are non-collapsed after measure adjustment.

Confirmed a conjecture by De Philippis and the second author.

Linked properties of Hessian trace and measure equivalence in RCD spaces.

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 $n$.