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<p>This paper concerns the evolution of a closed convex hypersurface in <InlineEquation ID="IEq1"> <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>, in direction of its inner unit normal vector, where the speed is given by a smooth function depending only on the mean curvature, and satisfies some further restrictions, without requiring homogeneity. It is shown that the flow exists on a finite maximal interval, convexity is preserved and the hypersurfaces shrink down to a single point as the final time is approached. This generalises the corresponding result of Schulze for the positive power mean curvature flow to a much larger possible class of flows by the functions depending only on the mean curvature.</p>

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Contracting Convex Hypersurfaces by Functions of the Mean Curvature

  • 期刊论文

  • 发表日期：

    2026年4月9日

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  • Shunzi Guo

摘要

This paper concerns the evolution of a closed convex hypersurface in \(\mathbb {R}^{n+1}\) R n + 1 , in direction of its inner unit normal vector, where the speed is given by a smooth function depending only on the mean curvature, and satisfies some further restrictions, without requiring homogeneity. It is shown that the flow exists on a finite maximal interval, convexity is preserved and the hypersurfaces shrink down to a single point as the final time is approached. This generalises the corresponding result of Schulze for the positive power mean curvature flow to a much larger possible class of flows by the functions depending only on the mean curvature.