TY - JOUR

T1 - Characterization of Tangent Cones of Noncollapsed Limits with Lower Ricci Bounds and Applications

AU - Colding, Tobias Holck

AU - Naber, Aaron

N1 - Funding Information:
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.
Copyright:
Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2013/2

Y1 - 2013/2

N2 - Consider a limit space (Mαgαpα →GH (Y,dY,p), where the Mαn 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 ΩY,p ⊆ MGH 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 ΩY,p. In particular, we give necessary and sufficient conditions for an open smooth family Ω≡ (X,gs) of closed manifolds to satisfy Ω =ΩY,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 (Yn, dY, p) with n ≥ 3 such that at p there exists for every 0 ≤ k ≤ n-2 a tangent cone at p of the form ℝk × C(Xn-k-1), where Xn-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 (Y5, dY, 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 ℂP2#ℂP̄2 while others are homeomorphic to cones over S4.

AB - Consider a limit space (Mαgαpα →GH (Y,dY,p), where the Mαn 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 ΩY,p ⊆ MGH 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 ΩY,p. In particular, we give necessary and sufficient conditions for an open smooth family Ω≡ (X,gs) of closed manifolds to satisfy Ω =ΩY,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 (Yn, dY, p) with n ≥ 3 such that at p there exists for every 0 ≤ k ≤ n-2 a tangent cone at p of the form ℝk × C(Xn-k-1), where Xn-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 (Y5, dY, 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 ℂP2#ℂP̄2 while others are homeomorphic to cones over S4.

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U2 - 10.1007/s00039-012-0202-7

DO - 10.1007/s00039-012-0202-7

M3 - Article

AN - SCOPUS:84875644315

VL - 23

SP - 134

EP - 148

JO - Geometric and Functional Analysis

JF - Geometric and Functional Analysis

SN - 1016-443X

IS - 1

ER -