Does there exist a smooth embedded open annulus S⊂R^3 with strictly negative Gaussian curvature and injective Gauss map N:S→S^2 that contains a simple closed asymptotic curve gamma:S^1→ S satisfying II(dot gamma,dot gamma)=0 and having zero normal self-linking number, i.e. operatornamelk(gamma,gamma_varepsilon)=0 for a sufficiently small normal push-off gamma_varepsilon of gamma?

Ghomi–Raffaelli Problem 1.1 on closed asymptotic curves

Does there exist a smooth embedded open annulus $S\subset\mathbb{R}^3$ with strictly negative Gaussian curvature and injective Gauss map $N:S\to\mathbb{S}^2$ that contains a simple closed asymptotic curve $\gamma:S^1\to S$ satisfying $\mathrm{II}(\dot\gamma,\dot\gamma)=0$ and having zero normal self-linking number, i.e. $\operatorname{lk}(\gamma,\gamma_\varepsilon)=0$ for a sufficiently small normal push-off $\gamma_\varepsilon$ of $\gamma$ ?

References

Primary source

arXiv

Additional references

A solution to a problem of Ghomi and Raffaelli on closed asymptotic curves — arXiv — Weiran Ding

differential-geometry surface-theory

Posted 1 day ago · edited 1 day ago

Progress summary

Refreshed 1 day ago

Claimed solved

A new preprint claims to construct the requested closed asymptotic curve, but the solution has not received independent mathematical verification.

Ghomi and Raffaelli ask whether a negatively curved embedded surface in $\mathbb{R}^3$ can carry a closed asymptotic curve with injective Gauss map and linking number $0$ . Their paper was submitted on December 26, 2024, and revised on September 10, 2025.

Known results

Ghomi and Raffaelli, 2024–2025: derived linking-number restrictions and constructed an example with injective Gauss map but linking number $2$ .
Ghomi and Raffaelli, 2024–2025: noted an earlier linking-number- $0$ example whose Gauss map is not injective.
Ghomi and Raffaelli, 2024–2025: showed that a negative answer would imply Nirenberg’s conjecture in the embedded setting.

October 6, 2026 claimed solution

Weiran Ding’s preprint claims an affirmative construction using rounded polygonal curves and a vanishing mixed integral, apparently supplying the missing example. The claim is currently unverified.

Current status (as of October 2026): A preprint claims the problem is solved, but independent verification is absent, so the mathematical resolution remains unconfirmed.

Sources

No solutions have been posted yet.