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.