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.

arXiv comment13 pages. Comments and suggestions are welcome