Collapsed Ricci limit spaces are non-collapsed RCD spaces

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Let (X,d,x,m)(X,\mathsf{d},x,\mathfrak{m}) be a collapsed Ricci limit space of Hausdorff dimension kk, with its limit measure satisfying

m=bHk\mathfrak{m}=b\mathcal{H}^k

for some b,k(0,)b,k\in(0,\infty). Collapsed Ricci limit space conjecture. If m=bHk\mathfrak{m}=b\mathcal{H}^k holds for some b,k(0,)b,k\in(0,\infty), then (X,d,Hk)(X,\mathsf{d},\mathcal{H}^k) is a non-collapsed RCD(K(n1),k)\operatorname{RCD}(K(n-1),k) space. This predicts that a collapsed Ricci limit space whose limit measure is proportional to Hausdorff measure has the synthetic Ricci-curvature structure of a non-collapsed RCD space; the source gives no resolution.

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Sources & referencesView supporting material

Primary source

Shouhei Honda, “Collapsed Ricci limit spaces as non-collapsed RCD spaces”, arXiv:2002.08612 (2020).

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