Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A mechanical characterization of CMC surfaces

Matteo Raffaelli

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.

A mechanical characterization of CMC surfaces

Abstract

The speed of a ball rolling without skidding or spinning on a surface is the length of the velocity of its center. We show that if the speed depends only on , then has constant mean curvature; and, conversely, that if the mean curvature of is constant and equal to , then either is a sphere or the ball of radius rolls on with direction-independent speed. It follows that the only surfaces where the speed is constant are subsets of planes, circular cylinders, and spheres.

Paper Structure

This paper contains 3 sections, 6 theorems, 15 equations.

Table of Contents

  1. Introduction
  2. Kinematic rolling
  3. Proof of the main result

Key Result

Theorem 1.1

If for some $r$ the initial speed of $B_{r}$ is the same for three pairwise nonparallel directions $v \in T_pS$, then $S$ is either umbilic or its mean curvature (with respect to $N$) equals $1/r$ at $p$. Conversely, if $S$ has mean curvature $h \neq 0$ and is umbilic (resp., nonumbilic) at $p$, the

Theorems & Definitions (14)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • proof
  • Remark 1.4
  • Corollary 1.5
  • Remark 1.6
  • Corollary 1.7
  • Theorem 2.1: nomizu1978raffaelli2018
  • Remark 2.2
  • ...and 4 more