Minimal Surfaces from Rigid Motions

TL;DR

This paper derives equations for hypersurfaces in higher dimensions that become minimal surfaces under rigid motions, revisiting classical cases with new insights into their geometric properties.

Contribution

It introduces new equations characterizing hypersurfaces that transform into minimal surfaces via rigid motions, extending classical results to higher dimensions.

Findings

Derived equations for hypersurfaces under rigid motions

Discussed classical minimal surface case N=2 in detail

Provided insights into geometric properties of these surfaces

Abstract

Equations are derived for the shape of a hypersurface in $\mathbb{R}^N$ for which a rigid motion yields a minimal surface in $\mathbb{R}^{N+1}$. Some elementary, but unconventional, aspects of the classical case $N=2$ (solved by H.F. Scherk in 1835) are discussed in some detail.