Notes on Ricci flows with collapsinginitial data (I): Distance distortion

Huang, Shaosai

doi:10.1090/tran/8034

Notes on Ricci flows with collapsinginitial data (I): Distance distortion
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by Shaosai Huang PDF

Trans. Amer. Math. Soc. 373 (2020), 4389-4414 Request permission

Abstract:

In this note, we prove a uniform distance distortion estimate for Ricci flows with uniformly bounded scalar curvature, independent of the lower bound of the initial $\boldsymbol {\mu }$-entropy. Our basic principle tells us that once correctly renormalized, the metric-measure quantities obey similar estimates as in the noncollapsing case; especially, the lower bound of the renormalized heat kernel, observed on a scale comparable to the initial diameter, matches with the lower bound of the renormalized volume ratio, giving the desired distance distortion estimate.

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