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<p>We consider rotationally symmetric, smooth, closed, convex hypersurfaces in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12220_2025_2040_Article_IEq3.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="34" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb {R}^{n+1}\)</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> on which we define a curvature flow that exhibits more than two distinct self-similar solutions. The pointwise speed of this flow is a non-symmetric function of the principal curvatures, without a weight factor dependent on the direction of the normal to the hypersurface at that point, making it, in this sense, an isotropic flow. However, its self-similar solutions are not necessarily round spheres even if the flow is also not affine invariant.</p>

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A Curvature Flow for Radially Symmetric Hypersurfaces in \({\mathbb {R}}^{n+1}\) by Non-symmetric Speed of their Principal Curvatures

  • 期刊论文

  • 发表日期：

    2025年5月24日

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  • Alina Stancu,
  • Valentina-Mira Wheeler

摘要

We consider rotationally symmetric, smooth, closed, convex hypersurfaces in \(\mathbb {R}^{n+1}\) R n + 1 on which we define a curvature flow that exhibits more than two distinct self-similar solutions. The pointwise speed of this flow is a non-symmetric function of the principal curvatures, without a weight factor dependent on the direction of the normal to the hypersurface at that point, making it, in this sense, an isotropic flow. However, its self-similar solutions are not necessarily round spheres even if the flow is also not affine invariant.