Geometric Analysis and Topology Seminar

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Convergence of Ricci flow and long-time existence of Harmonic map heat flow

Speaker: Yi Lai, University of California Irvine

Location: Warren Weaver Hall 512

Date: Friday, April 25, 2025, 11 a.m.

Synopsis:

For an ancient Ricci flow asymptotic to a compact integrable shrinker, or a Ricci flow developing a finite-time singularity modelled on the shrinker, we establish the long-time existence of a harmonic map heat flow between the Ricci flow and the shrinker for all times. This provides a global parabolic gauge for the Ricci flow and implies the uniqueness of the tangent flow without modulo any diffeomorphisms.

We present two main applications: First, we construct and classify all ancient Ricci flows asymptotic to any compact integrable shrinker, showing that they converge exponentially. Second, we obtain the optimal convergence rate at singularities modelled on such a shrinker, characterized by the first nontrivial stable eigenvalue of the stability operator for the entropy. This is joint work with K. Choi.