Characterization of Tangent Cones of Noncollapsed Limits with Lower Ricci Bounds and Applications
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Colding_Characterization of tangent.pdf
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Author(s)
Colding, Tobias
Naber, Aaron Charles
Date Issued
January 2013
Journal
Geometric and Functional Analysis
Publisher
Springer-Verlag
Citation
Colding, Tobias Holck, and Aaron Naber. “Characterization of Tangent Cones of Noncollapsed Limits with Lower Ricci Bounds and Applications.” Geometric and Functional Analysis 23, no. 1 (February 26, 2013): 134-148.
Version
Original manuscript
Abstract
Consider a limit space (M[subscript α],g[subscript α],p[subscript α]) [superscript GH over →] (Y, d[subscript Y], p), where the M[superscript n over α] 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 [superscript dash over Ω]Y,[subscript p] ⊆ M[subscript 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 [superscript dash over Ω]Y,[subscript p] . In particular, we give necessary and sufficient conditions for an open smooth family Ω ≡ (X, g[subscript s]) of closed manifolds to satisfy [superscript dash over Ω] = [superscript dash over Ω]Y,[subscript p] for some limit Y and point p ∈ Y as above, where [superscript dash over 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 [superscript 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[superscript k] × C(X[superscript n-k-1]), where X[superscript 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[superscript 5], d[subscript 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[superscript 2]#[superscript dash over CP][superscript 2] while others are homeomorphic to cones over S[superscript 4].
Description
Original manuscript January 6, 2012
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike 3.0
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DOI of Published Version
https://doi.org/10.1007/s00039-012-0202-7
