A conservation approach to helicoidal surfaces of constant mean curvature in R^3, S^3 and H^3

TL;DR

This paper introduces a conservation law framework for characterizing constant mean curvature helicoidal surfaces in three-dimensional spaces, linking geometric properties with differential equations and extending existing characterizations.

Contribution

It develops a conservation law for CMC surfaces using the `twizzler' construction and provides a converse characterization, connecting to known descriptions like the treadmillsled method.

Findings

CMC helicoidal surfaces are determined by a first-order ODE of the base curve.

Excluding cylinders, CMC surfaces can be characterized via conservation laws.

In R^3, the conservation law aligns with the treadmillsled characterization.

Abstract

We develop a conservation law for constant mean curvature (CMC) surfaces introduced by Korevaar, Kusner and Solomon, and provide a converse, so as to characterize CMC surfaces by a conservation law. We work with `twizzler' construction, which applies a screw-motion to some base curve. We show that, excluding cylinders, CMC helicoidal surfaces can be completely determined by a first-order ODE of the base curve. Further, we demonstrate that in R^3 this condition is equivalent to the treadmillsled characterization of helicoidal CMC surfaces given by O. Perdomo.