Perelman's entropy on ancient Ricci flows

Zilu Ma, Yongjia Zhang

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4 Scopus citations


In [33], the second author proved Perelman's assertion, namely, for an ancient Ricci flow with bounded and nonnegative curvature operator, bounded entropy is equivalent to noncollapsing on all scales. In this paper, we continue this discussion. It turns out that the curvature operator nonnegativity is not a necessary condition, and we need only to assume a consequence of Hamilton's trace Harnack. Furthermore, we show that this condition holds for steady Ricci solitons with nonnegative Ricci curvature.

Original languageEnglish (US)
Article number109195
JournalJournal of Functional Analysis
Issue number9
StatePublished - Nov 1 2021

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  • Ancient solution
  • Entropy
  • Ricci flow


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