New exotic examples of Ricci limit spaces

TL;DR

This paper constructs new Ricci limit spaces with diverse tangent cones, including different Euclidean dimensions, and demonstrates the existence of collapsed Ricci limit spaces with specified tangent cones, advancing understanding of Ricci limit space structures.

Contribution

It introduces novel examples of Ricci limit spaces with multiple tangent cone types and shows how to realize specific Riemannian cones as tangent cones in collapsed limits.

Findings

Existence of Ricci limit spaces with tangent cones of different Euclidean dimensions.

Construction of collapsed Ricci limit spaces with prescribed tangent cones.

Improvement over Menguy's earlier examples.

Abstract

For any integers $m\geqslant n\geqslant 3$, 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 $\mathrm{Ric}_{g_i}\geqslant(n_i-1)\geqslant 1$, 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 $x$.