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Structure theory of metric measure spaces with lower Ricci curvature bounds
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Mondino, Andrea and Naber, Aaron (2019) Structure theory of metric measure spaces with lower Ricci curvature bounds. Journal of the European Mathematical Society, 21 (6). pp. 1809-1854. doi:10.4171/JEMS/874 ISSN 1435-9855.
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Official URL: http://dx.doi.org/10.4171/JEMS/874
Abstract
We prove that a metric measure space (X,d,m) satisfying finite dimensional lower Ricci curvature bounds and whose Sobolev space W1,2 is Hilbert is rectifiable. That is, a RCD∗(K,N)-space is rectifiable, and in particular for m-a.e. point the tangent cone is unique and euclidean of dimension at most N. The proof is based on a maximal function argument combined with an original Almost Splitting Theorem via estimates on the gradient of the excess. We also show a sharp integral Abresh–Gromoll type inequality on the excess function and an Abresh–Gromoll-type inequality on the gradient of the excess. The argument is new even in the smooth setting
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): | Ricci flow, Mathematical optimization, Transportation problems (Programming), Geometry, Differential, Probabilities, Combinatorial analysis, Riemannian manifolds | ||||||||
Journal or Publication Title: | Journal of the European Mathematical Society | ||||||||
Publisher: | European Mathematical Society Publishing House | ||||||||
ISSN: | 1435-9855 | ||||||||
Official Date: | June 2019 | ||||||||
Dates: |
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Volume: | 21 | ||||||||
Number: | 6 | ||||||||
Page Range: | pp. 1809-1854 | ||||||||
DOI: | 10.4171/JEMS/874 | ||||||||
Status: | Peer Reviewed | ||||||||
Publication Status: | Published | ||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||
Date of first compliant deposit: | 20 June 2019 | ||||||||
Date of first compliant Open Access: | 21 June 2019 | ||||||||
Open Access Version: |
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