English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We prove a new kind of estimate that holds on any manifold with lower Riccibounds. It relates the geometry of two small balls with the same radius,potentially far apart, but centered in the interior of a common minimizinggeodesic. It reveals new, previously unknown, properties that all generalizedspaces with a lower Ricci curvature bound must have and it has a number ofapplications.  This new kind of estimate asserts that the geometry of small balls along anyminimizing geodesic changes in a H\"older continuous way with a constantdepending on the lower bound for the Ricci curvature, the dimension of themanifold, and the distance to the end points of the geodesic. We give examplesthat show that the H\"older exponent, along with essentially all the otherconsequences that we show follow from this estimate, are sharp. The unifiedtheme for all of these applications is convexity.  Among the applications is that the regular set is convex for anynon-collapsed limit of Einstein metrics. In the general case of potentiallycollapsed limits of manifolds with just a lower Ricci curvature bound we showthat the regular set is weakly convex and $a.e.$ convex, that is almost everypair of points can be connected by a minimizing geodesic whose interior iscontained in the regular set. We also show two conjectures of Cheeger-Colding.One of these asserts that the isometry group of any, even collapsed, limit ofmanifolds with a uniform lower Ricci curvature bound is a Lie group; the keypoint for this is to rule out small subgroups. The other asserts that thedimension of any limit space is the same everywhere. Finally, we show that aReifenberg type property holds for collapsed limits and discuss why thisindicate further regularity of manifolds and spaces with Ricci curvaturebounds.

Sharp H\"older continuity of tangent cones for spaces with a lower Ricci curvature bound and applications