Analysis of Type I singularities in the harmonic Ricci flow

In $\href{https://doi.org/10.4310/CAG.2011.v19.n5.a4}{[8]}$, Enders, Müller and Topping showed that any blow up sequence of a Type I Ricci flow near a singular point converges to a non-trivial gradient Ricci soliton, leading them to conclude that for such flows all reasonable definitions of singular points agree with each other. We prove the analogous result for the harmonic Ricci flow, generalizing in particular results of Guo, Huang and Phong $\href{https://dx.doi.org/10.4310/CAG.2018.v26.n3.a5}{[11]}$ and Shi $\href{https://doi.org/10.48550/arXiv.1309.5684}{[25]}$. In order to obtain our result, we develop refined compactness theorems, a new pseudolocality theorem, and a notion of reduced length and volume based at the singular time for the harmonic Ricci flow.

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Received 14 December 2018

Accepted 2 September 2021

Published 10 August 2024