PreprintCC0arXiv
A solution to a problem of Ghomi and Raffaelli on closed asymptotic curves
Abstract
We construct a smooth embedded open annulus in Euclidean three-space with strictly negative Gaussian curvature and injective Gauss map that contains a simple closed asymptotic curve of zero normal self-linking. This gives an affirmative answer to Problem 1.1 of Ghomi and Raffaelli. The construction starts with two simple planar curves whose tangent vectors are positive multiples of one another. Their mixed integral is made zero by rounding a pair of polygons with one free length parameter.
Related
- A counterexample to Nirenberg's conjecture on closed asymptotic curves
- Formal solution and strong asymptotic expansion of some initial value problem in two complex time variables, infinite order irregular singularities and Mahler transforms
- Regular closed gradient ranges and asymptotic gradient values
- Asymptotic Expansions of the Fundamental Solution for a Mixed Local-Nonlocal Operator
- A filtered time stepping scheme for curve shortening flow for open and closed curves
BibTeX
@misc{ding2026solution,
title = {{A solution to a problem of Ghomi and Raffaelli on closed asymptotic curves}},
author = {Weiran Ding},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08050},
eprint = {2610.08050},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08050},
note = {CC0},
}
RIS
TY - UNPB
TI - A solution to a problem of Ghomi and Raffaelli on closed asymptotic curves
AU - Weiran Ding
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08050
UR - https://arxiv.org/abs/2610.08050
AB - We construct a smooth embedded open annulus in Euclidean three-space with strictly negative Gaussian curvature and injective Gauss map that contains a simple closed asymptotic curve of zero normal self-linking. This gives an affirmative answer to Problem 1.1 of Ghomi and Raffaelli. The construction starts with two simple planar curves whose tangent vectors are positive multiples of one another. Their mixed integral is made zero by rounding a pair of polygons with one free length parameter.
ER -