Lower Ricci Curvature, Branching, and Bi-Lipschitz Structure of Uniform Reifenberg Spaces

We study here limit spaces $(M_\alpha,g_\alpha,p_\alpha)\stackrel{GH}{\rightarrow} (Y,d_Y,p)$, where the $M_\alpha$ 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 $\cR(Y)\subseteq Y$, called {\it regular} points, along with coverings by the almost regular points $\cap_\epsilon \cup_r\cR_{\epsilon,r}(Y)=\cR(Y)$ such that each of the {\it Reifenberg sets} $\cR_{\epsilon,r}(Y)$ is bi-H\"older 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 $\cR_{\epsilon,r}(Y)$ 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 $\RR^n$ but convergence to the tangent cones is at a uniform rate in $Y$, such that there exists no $C^{1,\beta}$ embeddings of $Y$ into Euclidean space for any $\beta>0$. Further, despite the strong tangential regularity of $Y$, there exists a point $y\in Y$ such that every pair of minimizing geodesics beginning at $y$ branches to any order at $y$. More specifically, given {\it any} two unit speed minimizing geodesics $\gamma_1$, $\gamma_2$ beginning at $y$ and {\it any} $0\leq \theta\leq \pi$, there exists a sequence $t_i\to 0$ such that the angle $\angle \gamma_1(t_i)y\gamma_2(t_i)$ converges to $\theta$.