Lower Ricci curvature, branching and the bilipschitz structure of uniform Reifenberg spaces

• Main Authors: Colding, Tobias (Contributor), Naber, Aaron Charles (Contributor)
• Other Authors: Massachusetts Institute of Technology. Department of Mathematics (Contributor)
• Format: Article
• Language: English
• Published: Elsevier, 2017-07-11T18:40:08Z.
• Subjects: Article
• Online Access: Get fulltext

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.
National Science Foundation (U.S.) (Grant DMS 0606629)
National Science Foundation (U.S.) (Grant DMS 1104392)
National Science Foundation (U.S.). Focused Research Group (Grant DMS 0854774)
National Science Foundation (U.S.). Graduate Research Fellowship Program