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Global regularity of three-dimensional Ricci limit spaces
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McLeod , Andrew D. and Topping, Peter (2022) Global regularity of three-dimensional Ricci limit spaces. Transactions of the American Mathematical Society, 9 . pp. 345-370. doi:10.1090/btran/47 ISSN 0002-9947.
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Official URL: https://doi.org/10.1090/btran/47
Abstract
Miles Simon and the second author, in their recent work [Geom. Topol. 25 (2021), pp. 913–948], established a local bi-Hölder correspondence between weakly noncollapsed Ricci limit spaces in three dimensions and smooth manifolds. In particular, any open ball of finite radius in such a limit space must be bi-Hölder homeomorphic to some open subset of a complete smooth Riemannian three-manifold. In this work we build on the technology from Simon and the second author in [J. Differential Geometry, to appear] and [Geom. Topol. 25, (2021), pp. 913–948] to improve this local correspondence to a global-local correspondence. That is, we construct a smooth three-manifold and prove that the entire (weakly) noncollapsed three-dimensional Ricci limit space is homeomorphic to via a globally-defined homeomorphism that is bi-Hölder once restricted to any compact subset. Here the bi-Hölder regularity is with respect to the distance on where is any smooth complete metric on .
A key step in our proof is the construction of local pyramid Ricci flows, existing on uniform regions of spacetime, that are inspired by Hochard’s partial Ricci flows in the paper by Raphaël Hochard [Short-time existence of the Ricci flow on complete, non-collapsed 3-manifolds with Ricci curvature bounded from below, https://arxiv.org/abs/1603.08726, 2016]. Suppose is a complete smooth pointed Riemannian three-manifold that is (weakly) noncollapsed and satisfies a lower Ricci bound. Then, given any we construct a smooth Ricci flow living on a subset of spacetime that contains, for each , a cylinder , where is dependent only on the Ricci lower bound, the (weakly) noncollapsed volume lower bound and the radius (in particular independent of ) and with the property that throughout .
Item Type: | Journal Article | |||||||||
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Subjects: | Q Science > QA Mathematics | |||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | |||||||||
Library of Congress Subject Headings (LCSH): | Manifolds (Mathematics), Homeomorphisms, Riemannian manifolds, Three-manifolds (Topology) | |||||||||
Journal or Publication Title: | Transactions of the American Mathematical Society | |||||||||
Publisher: | American Mathematical Society | |||||||||
ISSN: | 0002-9947 | |||||||||
Official Date: | 19 May 2022 | |||||||||
Dates: |
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Volume: | 9 | |||||||||
Page Range: | pp. 345-370 | |||||||||
DOI: | 10.1090/btran/47 | |||||||||
Status: | Peer Reviewed | |||||||||
Publication Status: | Published | |||||||||
Reuse Statement (publisher, data, author rights): | “First published in Transactions of the American Mathematical Society in [volume/issue number and year], published by the American Mathematical Society,” and the copyright notice in proper form must be placed on all copies. | |||||||||
Access rights to Published version: | Open Access (Creative Commons) | |||||||||
Copyright Holders: | © 2020 American Mathematical Society | |||||||||
Date of first compliant deposit: | 24 April 2020 | |||||||||
Date of first compliant Open Access: | 24 April 2020 | |||||||||
RIOXX Funder/Project Grant: |
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