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Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces

Miles Simon and Peter M Topping

Geometry & Topology 25 (2021) 913–948

Given a three-dimensional Riemannian manifold containing a ball with an explicit lower bound on its Ricci curvature and positive lower bound on its volume, we use Ricci flow to perturb the Riemannian metric on the interior to a nearby Riemannian metric still with such lower bounds on its Ricci curvature and volume, but additionally with uniform bounds on its full curvature tensor and all its derivatives. The new manifold is near to the old one not just in the Gromov–Hausdorff sense, but also in the sense that the distance function is uniformly close to what it was before, and additionally we have Hölder/Lipschitz equivalence of the old and new manifolds.

One consequence is that we obtain a local bi-Hölder correspondence between Ricci limit spaces in three dimensions and smooth manifolds. This is more than a complete resolution of the three-dimensional case of the conjecture of Anderson, Cheeger, Colding and Tian, describing how Ricci limit spaces in three dimensions must be homeomorphic to manifolds, and we obtain this in the most general, locally noncollapsed case. The proofs build on results and ideas from recent papers of Hochard and the current authors.

Ricci flow, Ricci limit spaces
Mathematical Subject Classification 2010
Primary: 35K40, 35K55, 53C23, 53C44, 58J35
Received: 16 September 2019
Revised: 2 May 2020
Accepted: 3 May 2020
Published: 27 April 2021
Proposed: Tobias H Colding
Seconded: John Lott, Gang Tian
Miles Simon
Institut für Analysis und Numerik
Universität Magdeburg
Peter M Topping
Mathematics Institute
University of Warwick
United Kingdom