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

Abstract

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.