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<p>In Smoczyk (Elem Math 60(2):57-65, 2005. <a href="https://doi.org/10.4171/EM/9">https://doi.org/10.4171/EM/9</a>) showed that the expansion of convex hypersurfaces by the reciprocal of the harmonic mean curvature gives rise to a <i>linear</i> second-order equation for the evolution of the support function, with corresponding representation formulae for solutions. In this article, we consider <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="28_2025_1067_Article_IEq1.gif" Format="GIF" Height="20" Rendition="HTML" Resolution="72" Type="Linedraw" Width="51" /> </InlineMediaObject> <EquationSource Format="TEX">\(L^2(d\sigma )\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mi>L</mi> <mn>2</mn> </msup> <mrow> <mo stretchy="false">(</mo> <mi>d</mi> <mi>σ</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>-gradient flows for a class of higher-order curvature functionals. These give rise to higher-order linear parabolic equations for which we derive similar representation formulae for their solutions. With suitable restrictions on the initial convex hypersurface, solutions exist for all time and converge exponentially fast in the smooth topology to spheres. We also consider evolution by similar flows that keep certain integrals fixed and those that evolve one convex hypersurface to another. In an Appendix, we give some related second-order results.</p>

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Representation formulae for higher-order linear curvature flows

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    2025年4月5日

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  • James A. McCoy

摘要

In Smoczyk (Elem Math 60(2):57-65, 2005. https://doi.org/10.4171/EM/9) showed that the expansion of convex hypersurfaces by the reciprocal of the harmonic mean curvature gives rise to a linear second-order equation for the evolution of the support function, with corresponding representation formulae for solutions. In this article, we consider \(L^2(d\sigma )\) L 2 ( d σ ) -gradient flows for a class of higher-order curvature functionals. These give rise to higher-order linear parabolic equations for which we derive similar representation formulae for their solutions. With suitable restrictions on the initial convex hypersurface, solutions exist for all time and converge exponentially fast in the smooth topology to spheres. We also consider evolution by similar flows that keep certain integrals fixed and those that evolve one convex hypersurface to another. In an Appendix, we give some related second-order results.