Weighted normalized curve shortening flow with applications in pseudo-Euclidean spaces

Abstract

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.

49 pages, 4 figures. The area-bivector monotonicity in Section 9.3 was discovered by ChatGPT. The ideas underlying examples in Section 9 and Appendix C originated from the authors, while most of the detailed arguments in these parts were developed with the assistance of ChatGPT. In the remaining parts of the paper, ChatGPT only assisted with checking details, writing, and typesetting