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On the scalar curvature blow up conjecture in Ricci flowGeometry/Topology
It is a basic fact that the Riemannian curvature becomes unbounded at every finite-time singularity of the Ricci flow. Sesum showed that, more precisely, even the Ricci curvature becomes unbounded at every such singularity. Whether the same can be said about the scalar curvature has since remained a conjecture, which has resisted several attempts of resolution.
In this talk, I will present a new result that partially confirms this conjecture in dimension 4 and motivates some interesting questions in 4 dimensional Ricci flow. Its proof relies on a combination of multi-scale arguments and Perelman's Harnack inequality on the conjugate heat equation. As a byproduct, we obtain an unconventional backwards pseudolocality theorem, which holds in any dimension.
This project is joint work with Qi Zhang.