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Area Inequalities for Embedded Disks Spanning Unknotted Curves

Joel Hass·Jeffrey C. Lagarias·William P. Thurston
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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 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.

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BibTeXRIS

Joel Hass, Jeffrey C. Lagarias, William P. Thurston. 2004-07-10. Area Inequalities for Embedded Disks Spanning Unknotted Curves. https://arxiv.org/abs/math/0306313

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