Strong spacelike curve convergence conjecture

For every closed smooth strong spacelike curve $\gamma_0\subset\mathbb{R}^{2,q}$ of index $1$ , the solution $\gamma(\cdot,t)$ of the usual curve-shortening flow exists up to a finite time $T$ and shrinks to a circular point: after a suitable translation and rescaling by factors tending to infinity as $t\nearrow T$ , the rescaled curves converge smoothly, in fact exponentially, to a centered unit circle contained in a spacelike $2$ -plane. The result is asserted in the cited work under additional geometric hypotheses, including that the curve admits a one-to-one convex projection onto a spacelike plane.

References

Primary source

arXiv

Additional references

Weighted normalized curve shortening flow with applications in pseudo-Euclidean spaces — arXiv — Baichuan Hu, Xiang Ma

Posted 2 weeks ago · edited 2 weeks ago

Progress summary

Refreshed 2 weeks ago

Claimed solved

A recent paper claims to settle the conjecture under specific geometric assumptions, while showing that lightlike degeneration is the alternative outside the convergent case.

The conjecture concerns the behavior under curve-shortening flow of closed strong spacelike curves of index $1$ in $\mathbb{R}^{2,q}$ . The claimed conclusion is convergence, after rescaling, to a circular point in a spacelike plane; the result is conditional on the geometric hypotheses stated in the paper.

Known results

Andrews and Zhou proved round-point convergence for compact spacelike-convex submanifolds with one spacelike codimension; the cited source says this includes closed strong spacelike curves of index $1$ in $\mathbb{R}^{2,q}$ .

Claimed proof; date not stated

Baichuan Hu and Xiang Ma claim in “Weighted normalized curve shortening flow with applications in pseudo-Euclidean spaces” that their Theorem 9.2 proves the strong-spacelike convergence conjecture in the index- $1$ setting. Their broader theorem gives a dichotomy: either the flow develops null degeneration, or it shrinks to a point and the rescaled curves converge smoothly and exponentially to a centered unit circle; an explicit null-degenerate example is also given. No independent verification or reported gap was found.

Current status (as of September 2026): the index- $1$ conjecture is claimed proved under the paper’s stated geometric hypotheses, but that claim remains unverified and broader cases remain open.

Sources

No solutions have been posted yet.