On Weak Super Ricci Flow through Neckpinch
In this article, we study the Ricci flow neckpinch in the context of metric measure spaces. We introduce the notion of a Ricci flow metric measure spacetime and of a weak (refined) super Ricci flow associated to convex cost functions (cost functions which are increasing convex functions of the dista...
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De Gruyter
2021
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oai:doaj.org-article:81274ec2bc4b4a40bd185574b4831e8a2021-12-05T14:10:38ZOn Weak Super Ricci Flow through Neckpinch2299-327410.1515/agms-2020-0123https://doaj.org/article/81274ec2bc4b4a40bd185574b4831e8a2021-07-01T00:00:00Zhttps://doi.org/10.1515/agms-2020-0123https://doaj.org/toc/2299-3274In this article, we study the Ricci flow neckpinch in the context of metric measure spaces. We introduce the notion of a Ricci flow metric measure spacetime and of a weak (refined) super Ricci flow associated to convex cost functions (cost functions which are increasing convex functions of the distance function). Our definition of a weak super Ricci flow is based on the coupled contraction property for suitably defined diffusions on maximal diffusion subspaces. In our main theorem, we show that if a non-degenerate spherical neckpinch can be continued beyond the singular time by a smooth forward evolution then the corresponding Ricci flow metric measure spacetime through the singularity is a weak super Ricci flow for a (and therefore for all) convex cost functions if and only if the single point pinching phenomenon holds at singular times; i.e., if singularities form on a finite number of totally geodesic hypersurfaces of the form {x} × 𝕊n. We also show the spacetime is a refined weak super Ricci flow if and only if the flow is a smooth Ricci flow with possibly singular final time.Lakzian SajjadMunn MichaelDe Gruyterarticlesuper ricci flowweak ricci flowmetric measure spaceneckpinch singularityoptimal transportationprimary: 53e20, 53c21secondary: 53c23, 49q22AnalysisQA299.6-433ENAnalysis and Geometry in Metric Spaces, Vol 9, Iss 1, Pp 120-159 (2021) |
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DOAJ |
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DOAJ |
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EN |
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super ricci flow weak ricci flow metric measure space neckpinch singularity optimal transportation primary: 53e20, 53c21 secondary: 53c23, 49q22 Analysis QA299.6-433 |
| spellingShingle |
super ricci flow weak ricci flow metric measure space neckpinch singularity optimal transportation primary: 53e20, 53c21 secondary: 53c23, 49q22 Analysis QA299.6-433 Lakzian Sajjad Munn Michael On Weak Super Ricci Flow through Neckpinch |
| description |
In this article, we study the Ricci flow neckpinch in the context of metric measure spaces. We introduce the notion of a Ricci flow metric measure spacetime and of a weak (refined) super Ricci flow associated to convex cost functions (cost functions which are increasing convex functions of the distance function). Our definition of a weak super Ricci flow is based on the coupled contraction property for suitably defined diffusions on maximal diffusion subspaces. In our main theorem, we show that if a non-degenerate spherical neckpinch can be continued beyond the singular time by a smooth forward evolution then the corresponding Ricci flow metric measure spacetime through the singularity is a weak super Ricci flow for a (and therefore for all) convex cost functions if and only if the single point pinching phenomenon holds at singular times; i.e., if singularities form on a finite number of totally geodesic hypersurfaces of the form {x} × 𝕊n. We also show the spacetime is a refined weak super Ricci flow if and only if the flow is a smooth Ricci flow with possibly singular final time. |
| format |
article |
| author |
Lakzian Sajjad Munn Michael |
| author_facet |
Lakzian Sajjad Munn Michael |
| author_sort |
Lakzian Sajjad |
| title |
On Weak Super Ricci Flow through Neckpinch |
| title_short |
On Weak Super Ricci Flow through Neckpinch |
| title_full |
On Weak Super Ricci Flow through Neckpinch |
| title_fullStr |
On Weak Super Ricci Flow through Neckpinch |
| title_full_unstemmed |
On Weak Super Ricci Flow through Neckpinch |
| title_sort |
on weak super ricci flow through neckpinch |
| publisher |
De Gruyter |
| publishDate |
2021 |
| url |
https://doaj.org/article/81274ec2bc4b4a40bd185574b4831e8a |
| work_keys_str_mv |
AT lakziansajjad onweaksuperricciflowthroughneckpinch AT munnmichael onweaksuperricciflowthroughneckpinch |
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