Entropy, noncollapsing, and a gap theorem for ancient solutions to the Ricci flow

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Abstract

In this paper we discuss the asymptotic entropy for ancient solutions to the Ricci flow. We prove a gap theorem for ancient solutions, which could be regarded as an entropy counterpart of Yokota’s work. In addition, we prove that under some assumptions on one time slice of a complete ancient solution with nonnegative curvature operator, finite asymptotic entropy implies κ-noncollapsing on all scales. This result is used by the author [21] to prove Perelman’s assertion that on an ancient solution to the Ricci flow with bounded nonnegative curvature operator, bounded entropy is equivalent to noncollapsing on all scales; see section 11 in [17].

Original languageEnglish (US)
Pages (from-to)501-524
Number of pages24
JournalCommunications in Analysis and Geometry
Volume29
Issue number2
DOIs
StatePublished - 2021

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