Examples of tangent cones of non-collapsed Ricci limit spaces

TL;DR

This paper constructs examples of non-collapsed Ricci limit spaces in all dimensions at least 5, where multiple different manifolds with core metrics appear as cross sections of tangent cones at the same point, demonstrating non-uniqueness.

Contribution

It extends previous work by providing examples in all dimensions ≥5 where various manifolds with core metrics appear as tangent cone cross sections at a single point.

Findings

Multiple manifold types can appear as tangent cone cross sections at the same point.

Non-uniqueness of tangent cone homeomorphism types is demonstrated in all dimensions ≥5.

Examples include manifolds admitting core metrics, related to positive Ricci curvature studies.

Abstract

We give new examples of manifolds that appear as cross sections of tangent cones of non-collapsed Ricci limit spaces. It was shown by Colding-Naber that the homeomorphism types of the tangent cones of a fixed point of such a space do not need to be unique. In fact, they constructed an example in dimension 5 where two different homeomorphism types appear at the same point. In this note, we extend this result and construct limit spaces in all dimensions at least 5 where any finite collection of manifolds that admit core metrics, a type of metric introduced by Perelman and Burdick to study Riemannian metrics of positive Ricci curvature on connected sums, can appear as cross sections of tangent cones of the same point.