Title:
Geometry at Infinity of Singularity Models of Ricci Flow
Abstract: Based on Hamilton's program, Perelman solved the
Poincaré conjecture and Thurston's geometrization conjecture using
Ricci flow with surgeries. It is crucial to understand the singularity
formation in order to perform surgeries. To capture the geometry in
the large of the singularity models, Perelman developed a space-time
comparison geometry, the L-geometry, to show the existence
of the asymptotic shrinkers, which is a smooth blow-down limit as time
goes to ∞. Despite the miraculous success in dimension 3,
Perelman's machinery is not suitable for higher dimensions partly due
to the complexity of curvatures. Recently, Richard Bamler developed a
compactness and partial regularity regularity theory for noncollapsed
Ricci flows, which greatly advanced the study of Ricci flows in
dimension greater or equal to 4. Among other important results, Bamler
introduced a notion of tangent flow at infinity, which is a blow-down
limit with respect to Bamler's new F-distance, and its
existence is canonical thanks to Bamler's compactness theory. In a
recent work with Chan and Zhang, roughly speaking, we proved that the
two notions of blow-downs coincide. If time permits, we also mention
the study of tangent flows at infinity of 4-dimensional steady Ricci
solitons and how the geometry at infinity determines the geometry in
the large, which is a recent work joint with Bamler, Chow, Deng and
Zhang.