On the rectifiability of $\mathsf{CD}(K,N)$ and $\mathsf{MCP}(K,N)$ spaces with unique tangents

Abstract: We prove rectifiability results for $\mathsf{CD}(K,N)$ and $\mathsf{MCP}(K,N)$ metric measure spaces $(\mathsf{X},\mathsf{d},\mathfrak{m})$ with pointwise Ahlfors regular reference measure $\mathfrak{m}$ and with $\mathfrak{m}$-almost everywhere unique metric tangents. In particular, we show rectifiability if (i) $(\mathsf{X},\mathsf{d},\mathfrak{m})$ is $\mathsf{CD}(K,N)$ for an arbitrary $N$ and has Hausdorff dimension $n<5$, or (ii) $(\mathsf{X},\mathsf{d},\mathfrak{m})$ is $\mathsf{MCP}(K,N)$ and non-collapsed, namely it has Hausdorff dimension $N$. Our strategy is based on the failure of the $\mathsf{CD}$ condition in sub-Finsler Carnot groups, on a new result on the failure of the non-collapsed $\mathsf{MCP}$ on sub-Finsler Carnot groups, and on the recent breakthrough by Bate [Invent. Math., 230(3):995-1070, 2022].