加权归一化曲线缩短流及其在伪欧几里得空间中的应用
Weighted normalized curve shortening flow with applications in pseudo-Euclidean spaces
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中文总结 AI 辅助
本文研究伪欧几里得空间中闭类空曲线的曲线缩短流，建立加权归一化流的二分法，证明曲线要么收缩为渐近圆点，要么切线趋于零锥，并给出面积-双向量缺陷作为点坍缩的定量障碍。
中文摘要 AI 辅助
本文关注伪欧几里得空间中闭类空曲线的曲线缩短流，目前这方面的结果很少。它们是否会产生某些切线趋于光锥的奇点？若不会，这样的曲线是否会收缩到一个圆点？为回答这些问题，我们对具有一致正且有界权重的平面加权归一化曲线缩短流建立了一个二分法。将其应用于伪欧几里得空间中闭光滑类空曲线（这些曲线允许到某个类空平面的一对一凸投影），在它们的有限最大时间处，我们将看到：要么曲线收缩到一个点并渐近地变成圆形，要么切线方向子序列趋于零锥。这两种情况都会发生。在第一种情况下，这证明了我们之前的猜想：在$\mathbb{R}^{2,q}$中指标为1的强类空曲线在通常的CSF下将收敛到一个圆点。在后一种情况下，在$\mathbb R^{2,1}$中发现了一个单调的面积-双向量缺陷，这给出了点坍缩的定量障碍。给出了具有类光切线极限的类空曲线的显式例子。
英文摘要
The focus of this paper is the curve shortening flow for closed spacelike curves in pseudo-Euclidean spaces, which has very few results so far. Will they produce singularities where certain tangent line tends to light cone? If not, will such a curve shrink to a circular point? To answer these questions, we establish a dichotomy for planar weighted normalized curve shortening flow with uniformly positive and bounded weights. Applying to closed smooth spacelike curves in pseudo-Euclidean spaces that admit a one-to-one convex projection onto a spacelike plane, at their finite maximal time we will see: either the curve shrinks to a point and becomes asymptotically circular, or the tangent directions subsequentially approach the null cone. Both alternatives occur. In the first case, this proves our previous conjecture that a strong spacelike curve in $\mathbb{R}^{2,q}$ with index 1 will converge to a circular point under the usual CSF. In the latter case, a monotone area-bivector defect is found in $\mathbb R^{2,1}$, which gives a quantitative obstruction to point collapse. Explicit examples of spacelike curves with lightlike tangent limit are given.
发表机构
- School of Mathematical Sciences, Peking University（北京大学数学科学学院）
- LMAM, School of Mathematical Sciences, Peking University（北京大学数学科学学院，应用数学研究所）
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