Distortion of spheres and surfaces in space

Abstract

It is known that the surface of a cone over the unit disc with large height has smaller distortion than the standard embedding of the 2-sphere in ${\mathbb{R}}^3$. In this note we show that distortion minimisers exist among convex embedded 2-spheres and have uniformly bounded eccentricity. Moreover, we prove that $\ddot{I}€/2$ is a sharp lower bound on the distortion of embedded closed surfaces of positive genus.

9 pages, 1 figure