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<p>Generalized Ricci flow is one of the natural extensions of Ricci flow, which appeared in the context of generalized geometry. It is known that the Perelman-type steady and expanding entropy functionals are monotone along generalized Ricci flow. In contrast to these two entropy functionals, the shrinking entropy functional is not monotone along generalized Ricci flow. The discovery of a monotone quantity fixed on shrinking solitons for generalized Ricci flow in full generality is an important open problem. We prove that the shrinking entropy functional is monotone along generalized Ricci flow with a perturbed term, and is fixed on a shrinking generalized Ricci soliton-type solution. Using the new monotonicity formula, we also prove a version of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1961_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\kappa \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>κ</mi> </math></EquationSource> </InlineEquation>-non-collapsing theorem for generalized Ricci flow with a perturbed term.</p>

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On the Shrinking Entropy Functional for Generalized Ricci Flow

  • 期刊论文

  • 发表日期：

    2025年3月28日

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  • Masashi Ishida

摘要

Generalized Ricci flow is one of the natural extensions of Ricci flow, which appeared in the context of generalized geometry. It is known that the Perelman-type steady and expanding entropy functionals are monotone along generalized Ricci flow. In contrast to these two entropy functionals, the shrinking entropy functional is not monotone along generalized Ricci flow. The discovery of a monotone quantity fixed on shrinking solitons for generalized Ricci flow in full generality is an important open problem. We prove that the shrinking entropy functional is monotone along generalized Ricci flow with a perturbed term, and is fixed on a shrinking generalized Ricci soliton-type solution. Using the new monotonicity formula, we also prove a version of \(\kappa \) κ -non-collapsing theorem for generalized Ricci flow with a perturbed term.