New exotic examples of Ricci limit spaces

Abstract

For any integers $m⩾n⩾3$, we construct a Ricci limit space $X_{m,n}$ such that for a fixed point, some tangent cones are ${\mathbb{R}}^m$ and some are ${\mathbb{R}}^n$. This is an improvement of Menguy's example. Moreover, we show that for any finite collection of closed Riemannian manifolds $(M_i^{n_i},g_i)$ with ${\mathrm{Ric}}_{g_i}⩾(n_i-1)⩾1$, there exists a collapsed Ricci limit space $(X,d,x)$ such that each Riemannian cone $C(M_i,g_i)$ is a tangent cone of $X$ at $x$.

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