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<p>We provide a collection of qualitative and quantitative dynamical aspects of the volume preserving mean curvature flow of closed smooth immersed curves in the plane including a lower bound for the existence time in terms of initial curvature variation and a sufficient condition for finite-time blow-up of curvature in the case of an immersed closed initial curve with positive signed area and rotation index <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_1994_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(+1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. We give a sufficient condition for immersed closed initial curves to become strictly locally convex and prove that initial curves with zero signed area produce a curvature blow-up.</p>

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On Finite Time Blow-Up of the Volume Preserving Mean Curvature Flow in 2D

  • 期刊论文

  • 发表日期：

    2025年4月11日

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  • Friedrich Lippoth

摘要

We provide a collection of qualitative and quantitative dynamical aspects of the volume preserving mean curvature flow of closed smooth immersed curves in the plane including a lower bound for the existence time in terms of initial curvature variation and a sufficient condition for finite-time blow-up of curvature in the case of an immersed closed initial curve with positive signed area and rotation index \(+1\) + 1 . We give a sufficient condition for immersed closed initial curves to become strictly locally convex and prove that initial curves with zero signed area produce a curvature blow-up.