A mechanical characterization of constant mean curvature surfaces

The speed of a ball rolling without skidding or spinning on a surface $S$ is the length of the velocity of its center.

We show that if the speed depends only on $p\in S$, then $S$ has constant mean curvature; and, conversely, that if the mean curvature of $S$ is constant and equal to $H>0$, then either $S$ is a sphere or the ball of radius $1/H$ rolls on $S$ with direction-independent speed.

It follows that the only surfaces where the speed is constant are subsets of planes, circular cylinders, and spheres.