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

We study here limit spaces (Mα,gα,pα)→GH(Y,dY,p), where the MαMα 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)⊆YR(Y)⊆Y, called regular   points, along with coverings by the almost regular points ⋂ϵ⋃rRϵ,r(Y)=R(Y)⋂ϵ⋃rRϵ,r(Y)=R(Y) such that each of the Reifenberg sets  Rϵ,r(Y)Rϵ,r(Y) is bihölder homeomorphic to a manifold. It has been an ongoing question as to the bilipschitz regularity the Reifenberg sets. Our results have two parts in this paper. First we show that each of the sets Rϵ,r(Y)Rϵ,r(Y) are bilipschitz embeddable into Euclidean space. Conversely, we show the bilipschitz 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 RnRn but convergence to the tangent cones is at a uniform rate in Y  , such that there exist no C1,βC1,β embeddings of Y   into Euclidean space for any β>0β>0. Further, despite the strong tangential regularity of Y  , there exists a point y∈Yy∈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 γ1γ1, γ2γ2 beginning at y and any  0⩽θ⩽π0⩽θ⩽π, there exists a sequence ti→0ti→0 such that the angle ∠γ1(ti)yγ2(ti)∠γ1(ti)yγ2(ti) converges to θ.