## Abstract

Consider a limit space (M_{α}g_{α}p_{α} →^{GH} (Y,d_{Y},p), where the M_{α}^{n} have a lower Ricci curvature bound and are volume noncollapsed. The tangent cones of Y at a point p ∈ Y are known to be metric cones C(X), however they need not be unique. Let Ω_{Y,p} ⊆ M_{GH} be the closed subset of compact metric spaces X which arise as cross sections for the tangents cones of Y at p. In this paper we study the properties of Ω_{Y,p}. In particular, we give necessary and sufficient conditions for an open smooth family Ω≡ (X,g_{s}) of closed manifolds to satisfy Ω =Ω_{Y,p} for some limit Y and point Ω is the closure of Ω in the set of metric spaces equipped with the Gromov-Hausdorff topology. We use this characterization to construct examples which exhibit fundamentally new behaviors. The first application is to construct limit spaces (Y^{n}, d_{Y}, p) with n ≥ 3 such that at p there exists for every 0 ≤ k ≤ n-2 a tangent cone at p of the form ℝ^{k} × C(X^{n-k-1}), where X^{n-k-1} is a smooth manifold not isometric to the standard sphere. In particular, this is the first example which shows that a stratification of a limit space Y based on the Euclidean behavior of tangent cones is not possible or even well defined. It is also the first example of a three dimensional limit space with nonunique tangent cones. The second application is to construct a limit space (Y^{5}, d_{Y}, p), such that at p the tangent cones are not only not unique, but not homeomorphic. Specifically, some tangent cones are homeomorphic to cones over ℂP^{2}#ℂP̄^{2} while others are homeomorphic to cones over S^{4}.

Original language | English (US) |
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Pages (from-to) | 134-148 |

Number of pages | 15 |

Journal | Geometric and Functional Analysis |

Volume | 23 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2013 |

### Funding

The first author was partially supported by NSF Grants DMS 0606629, DMS 1104392, and NSF FRG grant DMS 0854774 and the second author by an NSF Postdoctoral Fellowship.

## ASJC Scopus subject areas

- Analysis
- Geometry and Topology