AI 中文总结
本文研究(n+1)维光锥半内部超曲面的演化，探讨含主曲率初等对称多项式的变分问题性质，证明光锥内曲率型流永久存在且收敛至长度不变的圆。
AI 中文摘要
本文研究(n+1)维光锥半内部超曲面的演化过程，针对由r阶初等对称多项式S_r（即所有r个不同主曲率乘积的和）构成的光滑函数f(S₁,…,Sₙ)对应的变分问题，探讨其相关的若干基本性质；同时分析光锥内局部定义的曲率型流，证明其永久存在性及光滑收敛性，最终收敛至长度与初始曲线相等的圆。
英文摘要
In this paper, we investigate the evolutionary processes of hypersurfaces within half of the $(n+1)$-dimensional light-cone. Depending on the evolutionary processes, our focus extends to exploring variational problems associated with a smooth function $f(S_1,\cdots,S_n)$, where each $S_r$ denotes the $r$-th elementary symmetric polynomial, defined as the sum of all possible products of $r$ distinct principal curvatures. We present several fundamental properties related to these variational problems. Furthermore, we examine a curvature-type flow defined locally within the light-cone, establishing its perpetual existence and smooth convergence to a circle whose length is preserved and equal to that of the initial curve.
