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<p>We prove that a complete solution to the Ricci flow on <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3234_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="91" /> </InlineMediaObject> <EquationSource Format="TEX">\(M\times [-T, 0)\)</EquationSource> </InlineEquation> which has quadratic curvature decay on some end <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3234_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="57" /> </InlineMediaObject> <EquationSource Format="TEX">\(E\subset M\)</EquationSource> </InlineEquation> and converges locally smoothly to the end of a cone on <i>E</i> as <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="208_2025_3234_Article_IEq3.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(t\nearrow 0\)</EquationSource> </InlineEquation> must be a gradient shrinking soliton.</p>

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Ricci flows which terminate in cones

  • 期刊论文

  • 发表日期：

    2025年8月1日

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  • Brett Kotschwar

摘要

We prove that a complete solution to the Ricci flow on \(M\times [-T, 0)\) which has quadratic curvature decay on some end \(E\subset M\) and converges locally smoothly to the end of a cone on E as \(t\nearrow 0\) must be a gradient shrinking soliton.