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PreprintarXiv licencearXiv

A counterexample to Nirenberg's conjecture on closed asymptotic curves

  • Harrison Banks
  • Mohammad Ghomi
  • Hongda Qiu
  • Matteo Raffaelli
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Abstract

We construct a smooth embedded annulus in Euclidean 3-space with negative Gauss curvature in its interior whose boundary consists of strictly convex curves in planes tangent to the annulus, and which contains closed asymptotic curves. This gives a counterexample to a 1962 conjecture of Nirenberg. The construction is based on Lelieuvre's formula, which yields a closed curve that is then realized as an asymptotic curve on a ruled annulus. The second asymptotic curve is obtained from a periodic solution of a Riccati equation.

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Cite

BibTeXRIS
BibTeX
@misc{banks2026counterexample,
  title = {{A counterexample to Nirenberg's conjecture on closed asymptotic curves}},
  author = {Harrison Banks and Mohammad Ghomi and Hongda Qiu and Matteo Raffaelli},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.13848},
  eprint = {2610.13848},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.13848},
}
RIS
TY  - UNPB
TI  - A counterexample to Nirenberg's conjecture on closed asymptotic curves
AU  - Harrison Banks
AU  - Mohammad Ghomi
AU  - Hongda Qiu
AU  - Matteo Raffaelli
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.13848
UR  - https://arxiv.org/abs/2610.13848
AB  - We construct a smooth embedded annulus in Euclidean 3-space with negative Gauss curvature in its interior whose boundary consists of strictly convex curves in planes tangent to the annulus, and which contains closed asymptotic curves. This gives a counterexample to a 1962 conjecture of Nirenberg. The construction is based on Lelieuvre's formula, which yields a closed curve that is then realized as an asymptotic curve on a ruled annulus. The second asymptotic curve is obtained from a periodic solution of a Riccati equation.
ER  -