| 田亚,方建波.关于平面浸入星形曲线的一类演化问题[J].数学年刊A辑,2026,(1):33~44 |
| 关于平面浸入星形曲线的一类演化问题 |
| A Class of Evolution Problems for Planar Immersed Star-Shaped Curves |
| Received:December 13, 2025 Revised:March 04, 2026 |
| DOI：10.16205/j.cnki.cama.2026.0003 |
| 中文关键词: 局部星形曲线 Yau问题 径向函数 抛物方程 |
| 英文关键词:Locally star-shaped curves Yau's problem Radial function Parabolic equation |
| 基金项目:国家自然科学基金(No.12561011);贵州省科技厅项目: 黔科合基础(No.MS(2026)071) |
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| 中文摘要: |
| 受[Gao L Y, Wang Z Q. Evolving an immersed star-shaped curve to another one with same winding number [J]. J Differential Equations, 2025, 424:421–437]工作的启发, 对于平面浸入星形曲线, 引入了一种能量E(t)=∫mS11/(rln(r+1))dθ, 并通过构造一类含有非局部项的抛物型曲线流, 得到了在这个流下, 演化曲线能保持星形性及上述能量守恒; 同时, 当时间t趋于无穷时, 演化曲线会光滑收敛到与它具有相同缠绕数的星形曲线. 这一结果回答了Yau在2007年提出的一个问题. |
| 英文摘要: |
| Inspired by the work of [Gao L Y, Wang Z Q. Evolving an immersed star-shaped curve to another one with same winding number [J]. J Differential Equations, 2025, 424:421–437], the authors introduce an energy functional E(t)=∫mS11/(rln(r+1))dθ for planar immersed star-shaped curves. By constructing a parabolic curve flow involving a nonlocal term, the authors show that under this flow the evolving curves preserve star-shapedness and the energy remains constant. Moreover, as time t tends to infinity, the evolving curve converges smoothly to a star-shaped curve with the same winding number as the initial curve. This result answers a question posed by Yau in 2007. |
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