Constant mean curvature surfaces via integrable dynamical system

TL;DR

This paper demonstrates that the equation governing constant mean curvature surfaces can be expressed as a Hamiltonian system, with its finite-dimensional reduction yielding integrable trajectories corresponding to classical Delaunay and do Carmo-Dajzcer surfaces.

Contribution

It introduces a Hamiltonian formulation for the constant mean curvature surface equation and connects finite-dimensional reductions to known integrable surfaces.

Findings

The governing equation has Hamiltonian form.

Finite-dimensional reduction is integrable.

Trajectories correspond to classical CMC surfaces.

Abstract

It is shown that the equation which describes constant mean curvature surface via the generalized Weierstrass-Enneper inducing has Hamiltonian form. Its simplest finite-dimensional reduction has two degrees of freedom, integrable and its trajectories correspond to well-known Delaunay and do Carmo-Dajzcer surfaces (i.e., helicoidal constant mean curvature surfaces).