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Paper   IPM / M / 16707
School of Mathematics
  Title:   On weak super Ricci flow through neckpinch
  Author(s):  Sajjad Lakzian (Joint with M. Munn)
  Status:   Published
  Journal: Anal. Geom. Metr. Spaces
  Vol.:  9
  Year:  2021
  Pages:   120-159
  Supported by:  IPM
n 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 components. 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} ×\spheren. 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.

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