Sharp Holder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications
Name
Colding_Sharp holder.pdf
Size
388.66 KB
Format
Adobe PDF
Checksum (MD5)
0a25ee0eff814b9df5128963dfb02dd8
Author(s)
Colding, Tobias
Naber, Aaron Charles
Date Issued
September 2012
Journal
Annals of Mathematics
Publisher
Princeton University Press
Citation
Colding, Tobias, and Aaron Naber. “Sharp Hölder continuity of tangent cones for spaces with a lower Ricci curvature bound and applications.” Annals of Mathematics 176, no. 2 (September 1, 2012): 1173-1229.
Version
Original manuscript
Abstract
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 noncollapsed 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.
Description
Original manuscript September 22, 2011
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
Terms of Use
Creative Commons Attribution-Noncommercial-Share Alike 3.0
Persistent DSpace Link
DOI of Published Version
https://doi.org/10.4007/annals.2012.176.2.10
