Area inequalities for embedded disks spanning unknotted curves
| Venue: | 2003, arXiv:math.DG/0306313. EFFICIENTLY BOUND 4-MANIFOLDS 43 |
| Citations: | 4 - 1 self |
BibTeX
@INPROCEEDINGS{Hass_areainequalities,
author = {Joel Hass and Jeffrey C. Lagarias and William P. Thurston},
title = {Area inequalities for embedded disks spanning unknotted curves},
booktitle = {2003, arXiv:math.DG/0306313. EFFICIENTLY BOUND 4-MANIFOLDS 43},
year = {}
}
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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→0for 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. 1
