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- W4300921142 abstract "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 $cap_epsilon cup_rcR_{epsilon,r}(Y)=cR(Y)$ such that each of the {it Reifenberg sets} $cR_{epsilon,r}(Y)$ is bi-Holder 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 $yin Y$ 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} $0leq thetaleq pi$, there exists a sequence $t_ito 0$ such that the angle $angle gamma_1(t_i)ygamma_2(t_i)$ converges to $theta$." @default.
- W4300921142 created "2022-10-04" @default.
- W4300921142 creator A5014724625 @default.
- W4300921142 creator A5017917162 @default.
- W4300921142 date "2011-11-09" @default.
- W4300921142 modified "2023-09-26" @default.
- W4300921142 title "Lower Ricci Curvature, Branching, and Bi-Lipschitz Structure of Uniform Reifenberg Spaces" @default.
- W4300921142 doi "https://doi.org/10.48550/arxiv.1111.2184" @default.
- W4300921142 hasPublicationYear "2011" @default.
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