|Location:||MSRI: Simons Auditorium|
The formation of neckpinch singularities is a familiar feature of both Ricci flow and mean curvature flow: They occur in either flow if the initial geometry or initial embedding includes sufficiently tight corseting of a neck which connects a pair of less tightly corseted regions. The neckpinches are called nondegenerate if the corresponding singularity model is cylindrical, and degenerate otherwise. One expects nondegenerate neckpinches (and Type I singularities) if the initial neck corseting is very tight, while degenerate neckpinches (and Type II singularities) are expected if the initial neck corseting tightness is at a threshold value, between those which lead to singularity formation and those which do not.
We focus on two issues concerning neckpinch singularities: First, we discuss the apparent relationship between topology and the rate of curvature blowup in degenerate neckpinches. Recent work supports the conjecture that both in mean curvature flow and in Ricci flow, if the manifold is closed one has discrete rates of curvature blowup, while if the manifold is not closed then a continuum of possible rates of curvature blowup can occur. The evidence for this is based on studies of rotationally-symmetric flows. The second issue we discuss is the tendency of neckpinches to approach maximal symmetry. Recent work indicates that neckpinch-forming flows which are either nearly rotationally-symmetric or are deformed from rotational symmetry in certain ways will asymptotically approach rotational symmetry, at least in a neighborhood of the neckpinch.