Area Inequalities for Embedded Disks Spanning Unknotted Curves
Abstract
We show that a smooth unknotted curve in R^3 satisfies an isoperimetric inequality that bounds the area of an embedded disk spanning the curve in terms of two parameters: the length L of the curve and the thickness r (maximal radius of an embedded tubular neighborhood) of the curve. For fixed length, the expression giving the upper bound on the area grows exponentially in 1/r^2. In the direction of lower bounds, we give a sequence of length one curves with r approaching 0 for which the area of any spanning disk is bounded from below by a function that grows exponentially with 1/r. In particular, given any constant A, there is a smooth, unknotted length one curve for which the area of a smallest embedded spanning disk is greater than A.
Cite
@article{arxiv.math/0306313,
title = {Area Inequalities for Embedded Disks Spanning Unknotted Curves},
author = {Joel Hass and Jeffrey C. Lagarias and William P. Thurston},
journal= {arXiv preprint arXiv:math/0306313},
year = {2007}
}
Comments
31 pages, 5 figures