TY - JOUR

T1 - Lower Ricci curvature, branching and the bilipschitz structure of uniform Reifenberg spaces

AU - Colding, Tobias Holck

AU - Naber, Aaron

N1 - Funding Information:
The first author was partially supported by NSF grant DMS 0606629 and NSF FRG grant DMS 0854774 and the second author by an NSF Postdoctoral Fellowship .

PY - 2013/12/20

Y1 - 2013/12/20

N2 - We study here limit spaces (Mα,gα,pα)→GH(Y,dY,p), where the Mα 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 n-ary intersectionεn-ary unionrRε,r(Y)=R(Y) such that each of the Reifenberg sets 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) 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 Rn but convergence to the tangent cones is at a uniform rate in Y, such that there exist no C1,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 γ1, γ2 beginning at y and any 0 ≤ θ ≤ π, there exists a sequence t i → 0 such that the angle ∠γ1(t i)yγ2(t i) converges to θ.

AB - We study here limit spaces (Mα,gα,pα)→GH(Y,dY,p), where the Mα 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 n-ary intersectionεn-ary unionrRε,r(Y)=R(Y) such that each of the Reifenberg sets 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) 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 Rn but convergence to the tangent cones is at a uniform rate in Y, such that there exist no C1,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 γ1, γ2 beginning at y and any 0 ≤ θ ≤ π, there exists a sequence t i → 0 such that the angle ∠γ1(t i)yγ2(t i) converges to θ.

KW - Bilipschitz

KW - Branching

KW - Reifenberg spaces

KW - Ricci curvature

UR - http://www.scopus.com/inward/record.url?scp=84885360094&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84885360094&partnerID=8YFLogxK

U2 - 10.1016/j.aim.2013.09.005

DO - 10.1016/j.aim.2013.09.005

M3 - Article

AN - SCOPUS:84885360094

VL - 249

SP - 348

EP - 358

JO - Advances in Mathematics

JF - Advances in Mathematics

SN - 0001-8708

ER -