Sharp Holder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications

By T. Colding, A. Naber. 2011

248 citations recorded by the source corpus.

We prove a new estimate on manifolds with a lower Ricci bound which asserts that the geometry of balls centered on a minimizing geodesic can change in at most a Holder continuous way along the geodesic. We give examples that show that the Holder exponent, along with essentially all the other consequences that follow from this estimate, are sharp. Among the applications is that the regular set is convex for any non- collapsed limit of Einstein metrics. In the general case of a potentially collapsed limit of manifolds with just a lower Ricci curvature bound we show that the regular set is weakly convex and a.e. convex. We also show two conjectures of Cheeger-Colding. One of these asserts that the isometry group of any, even collapsed, limit of manifolds with a uniform lower Ricci curvature bound is a Lie group. The other asserts that the dimension of any limit space is the same everywhere.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/228010ac51e3cb5f8e4a5c3aaa3817e919931ab7

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Citations in the corpus

On the volume measure of non-smooth spaces with Ricci curvature bounded below (2016, background): “Naber [16] that the dimension is independent of the chart.”
Integral bounds on curvature and Gromov–Hausdorff limits (2012, background): “C(Y) connecting pand qmust be of the form (r(t),γ(t)) where γ(t) is a geodesic in Y, and ris a universal function of dY (p,q) and tdetermined by elementary trigonometry. Byrecentresult ofColding-Naber[7] weknowC(Yreg) isgeodesically convex in C(Y), so the lemma follows. As usual there is a Reeb ﬁeld”
Sharp geometric and functional inequalities in metric measure spaces with lower Ricci curvature bounds (2015, background): “K;N / spaces are (weighted) Riemannian manifolds satisfying lower Ricci curvature bounds and their measured Gromov Hausdorff limits (for the theory of Ricci limit spaces, see Cheeger and Colding [20; 21; 22; 23] and Colding and Naber [24]), Alexandrov spaces satisfying lower curvature bounds and, mo”
Space of Ricci Flows I (2009, background): “Interested readers are referred to [24] for the most general version.”
Nonsmooth differential geometry - An approach tailored for spaces with Ricci curvature bounded from below (2014, result): “By analogy with the Ricci-limit case and from the paper [23] of Colding-Naber one expects the answer to be positive, but with the current technology the replication of the arguments in [23] in the non-smooth setting still presents some difficulties.”
K‐Stability and Kähler‐Einstein Metrics (2012, result): “Note that the claim can also be deduced from a result of Colding and Naber [17], who proved the convexity of M1 n S.”
New formulas for the Laplacian of distance functions and applications (2018, background): “Since those classical works, the Laplacian comparison Theorem has become a fundamental technical tool in the investigation of Riemannian manifolds satisfying Ricci curvature lower bounds (see for instance [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 41, 50]).”
Euclidean spaces as weak tangents of infinitesimally Hilbertian metric measure spaces with Ricci curvature bounded below (2013, background): “Only much later, in [16] Colding and Naber showed that in fact for any Ricci-limit space there exists k 2 N such that at almost every point in the space the tangent cone is R .”
Short-time behavior of the heat kernel and Weyl's law on $RCD^*(K, N)$-spaces (2017, background): “This property is known to be true for Ricci limit spaces, see [16].”