Area Inequalities for Embedded Disks Spanning Unknotted Curves

Details Area Inequalities for Embedded Disks Spanning Unknotted Curves Area Inequalities for Embedded Disks Spanning Unknotted Curves

2003-06-21

arxiv.org/abs/math/0306313v2

Journal of Differential Geometry 68, (2004) 1-30.

Mathematics

Differential Geometry

31 pages, 5 figures

Scientific paper

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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