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 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 ℝ^2,1, which gives a quantitative obstruction to point collapse. Explicit examples of spacelike curves with lightlike tangent limit are given.

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BibTeX RIS

BibTeX

@misc{hu2026weighted,
  title = {{Weighted normalized curve shortening flow with applications in pseudo-Euclidean spaces}},
  author = {Baichuan Hu and Xiang Ma},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.24027},
  eprint = {2609.24027},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.24027},
}

RIS

TY  - UNPB
TI  - Weighted normalized curve shortening flow with applications in pseudo-Euclidean spaces
AU  - Baichuan Hu
AU  - Xiang Ma
PY  - 2026
DA  - 2026/09/21
PB  - arXiv
DO  - 10.48550/arxiv.2609.24027
UR  - https://arxiv.org/abs/2609.24027
AB  - 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 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 ℝ^2,1, which gives a quantitative obstruction to point collapse. Explicit examples of spacelike curves with lightlike tangent limit are given.
ER  -

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