Area Inequalities for Embedded Disks Spanning Unknotted Curves

Area Inequalities for Embedded Disks Spanning Unknotted Curves

We show that a smooth unknotted curve in R3 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/r2. In the direction ...

The Size of Spanning Disks for Polygonal Curves

For each integer n ≥ 0, there is a closed, unknotted, polygonal curve Kn in R3 having less than 10n + 9 edges, with the property that any Piecewise-Linear triangulated disk spanning the curve contains at least 2n−1 triangles.

A Class of Unknotted Curves in 3-space

The size of spanning disks

For each integer n > 1 we construct a closed unknotted PL curve Kn in R 3 having less than 33n edges with the property that any PL triangluated disk spanning the curve contains at least 2 n triangles.

Relative isoperimetric inequalities for Lagrangian disks

for some universal constant μ by constructing an explicit isotropic deformation of C to a single point. In this note, we wish to prove a relative analogue of this result. Our interest comes from the application of the relative isoperimetric inequality to the regularity of an area minimizing Lagrangian surface [Wa]. Let ̟ = dx∧dy+dx∧dy be the standard symplectic form on C with coordinates z = x +...