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|a Colding, Tobias
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|a Massachusetts Institute of Technology. Department of Mathematics
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|a Colding, Tobias
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|a Naber, Aaron Charles
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|a Naber, Aaron Charles
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|a Lower Ricci curvature, branching and the bilipschitz structure of uniform Reifenberg spaces
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|b Elsevier,
|c 2017-07-11T18:40:08Z.
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|z Get fulltext
|u http://hdl.handle.net/1721.1/110645
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|a We study here limit spaces (M[subscript α], g[subscript α], p[subscript α]) [GH over →] (Y, d[subscript Y], p), where the M[subscript α] have a lower Ricci curvature bound and are volume noncollapsed. Such limits Y may be quite singular, however it is known that there is a subset of full measure R(Y) ⊆ Y, called regular points, along with coverings by the almost regular points ∩[subscript ε]∪[subscript r] R[subscript ε,r](Y) = R(Y) such that each of the Reifenberg sets R[subscript ε,r](Y) is bi-Hölder homeomorphic to a manifold. It has been an ongoing question as to the bi-Lipschitz regularity the Reifenberg sets.Our results have two parts in this paper. First we show that each of the sets R[subscript ε,r](Y) are bi-Lipschitz embeddable into Euclidean space. Conversely, we show the bi-Lipschitz nature of the embedding is sharp. In fact, we construct a limit space Y which is even uniformly Reifenberg, that is, not only is each tangent cone of Y isometric to R[superscript n] but convergence to the tangent cones is at a uniform rate in Y, such that there exists no C[superscript 1,β] embeddings of Y into Euclidean space for any β > 0. Further, despite the strong tangential regularity of Y, there exists a point y ∈ Y such that every pair of minimizing geodesics beginning at y branches to any order at y. More specifically, given any two unit speed minimizing geodesics γ[superscript 1], γ[subscript 2] beginning at y and any 0 ≤ θ ≤ π, there exists a sequence t[subscript i] → 0 such that the angle ∠ γ[subscript 1](t[subscript i])yγ[subscript 2](t[subscript i]) converges to θ.
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|a National Science Foundation (U.S.) (Grant DMS 0606629)
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|a National Science Foundation (U.S.) (Grant DMS 1104392)
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|a National Science Foundation (U.S.). Focused Research Group (Grant DMS 0854774)
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|a National Science Foundation (U.S.). Graduate Research Fellowship Program
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|a en_US
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|a Article
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|t Advances in Mathematics
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