A mechanical characterization of CMC surfaces

TL;DR

This paper explores the relationship between the speed of a rolling ball and the geometry of the surface, establishing that constant speed surfaces are planes, cylinders, or spheres, and linking speed dependence to constant mean curvature surfaces.

Contribution

It demonstrates a novel connection between the speed of a rolling ball and the constant mean curvature property of surfaces, providing a geometric characterization.

Findings

Surfaces with speed depending only on position have constant mean curvature.

Constant speed surfaces are limited to planes, cylinders, and spheres.

If mean curvature is non-zero and constant, the surface is either a sphere or a specific rolling surface.

Abstract

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 $H\neq 0$, 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.