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<p>If we want to deform a compact Riemannian manifold with boundary using Ricci flow, we first need to decide on appropriate boundary conditions. We would like these conditions to reflect the geometric nature of the flow and allow for a variety of initial data. Importantly, the conditions should be compatible with the expected evolution of Einstein metrics. We propose it’s natural to choose those conditions, for which the first variation of certain functionals, such as the Einstein-Hilbert action and Perelman’s <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2181_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="15" /> </InlineMediaObject> <EquationSource Format="TEX">\(\lambda \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>λ</mi> </math></EquationSource> </InlineEquation>-functional, does not admit a boundary term. We provide a proof of the short term existence of solutions of the initial boundary value problem, under these conditions.</p>

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Novel Boundary Conditions for the Ricci Flow

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月16日

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  • Rasmus Jouttijärvi

摘要

If we want to deform a compact Riemannian manifold with boundary using Ricci flow, we first need to decide on appropriate boundary conditions. We would like these conditions to reflect the geometric nature of the flow and allow for a variety of initial data. Importantly, the conditions should be compatible with the expected evolution of Einstein metrics. We propose it’s natural to choose those conditions, for which the first variation of certain functionals, such as the Einstein-Hilbert action and Perelman’s \(\lambda \) λ -functional, does not admit a boundary term. We provide a proof of the short term existence of solutions of the initial boundary value problem, under these conditions.