This paper focuses on the evolution problem for nonsimple curves with rotation number <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$m\in \mathbb {Z}^+$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation>. Motivated by length-preserving flows introduced by Pan-Yang (Manuscr Math 127:469–484, 2008) and Wang (J Funct Anal 284:109744, 2023), a locally constrained inverse curvature flow is considered. This flow exists in time interval <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$[0,+\infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">[</mo> <mn>0</mn> <mo>,</mo> <mo>+</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>, and under this flow, any locally convex curve of rotation number <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$m\in \mathbb {Z}^+$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>∈</mo> <msup> <mrow> <mi mathvariant="double-struck">Z</mi> </mrow> <mo>+</mo> </msup> </mrow> </math></EquationSource> </InlineEquation> maintains its length and deforms into an m-fold circle of center the origin as time t goes to infinity. As applications of this flow, the isoperimetric inequality and curvature-type inequalities are obtained for locally convex curves that is of total curvature of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$2m\pi $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>m</mi> <mi>π</mi> </mrow> </math></EquationSource> </InlineEquation> and n-fold symmetry (<InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\frac{m}{n}\le 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mfrac> <mi>m</mi> <mi>n</mi> </mfrac> <mo>≤</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>).

Evolution of nonsimple curves in the length-preserving locally constrained inverse curvature flow

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发表日期：

2025年6月27日

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Li Gao,
Mengrui Li,
Yunlong Yang

摘要

This paper focuses on the evolution problem for nonsimple curves with rotation number $m\in \mathbb {Z}^+$ $m \in Z^{+}$ . Motivated by length-preserving flows introduced by Pan-Yang (Manuscr Math 127:469–484, 2008) and Wang (J Funct Anal 284:109744, 2023), a locally constrained inverse curvature flow is considered. This flow exists in time interval $[0,+\infty )$ $[0, + \infty)$ , and under this flow, any locally convex curve of rotation number $m\in \mathbb {Z}^+$ $m \in Z^{+}$ maintains its length and deforms into an m-fold circle of center the origin as time t goes to infinity. As applications of this flow, the isoperimetric inequality and curvature-type inequalities are obtained for locally convex curves that is of total curvature of $2m\pi $ $2 m π$ and n-fold symmetry ( $\frac{m}{n}\le 1$ $\frac{m}{n} \leq 1$ ).