Polynomial conserved quantities for constrained Willmore surfaces

TL;DR

This paper introduces a hierarchy of constrained Willmore surfaces characterized by polynomial conserved quantities, linking them to known surface classes and demonstrating their invariance under certain transformations.

Contribution

It defines a new hierarchy of constrained Willmore surfaces based on polynomial conserved quantities and shows their invariance under spectral deformation and Baecklund transformation.

Findings

Hierarchy preserves under spectral deformation

Hierarchy preserved under Baecklund transformation

Transforms constant mean curvature surfaces into new ones

Abstract

We define a hierarchy of special classes of constrained Willmore surfaces by means of the existence of a polynomial conserved quantity of some type, filtered by an integer. Type 1 with parallel top term characterises parallel mean curvature surfaces and, in codimension 1, type 1 characterises constant mean curvature surfaces. We show that this hierarchy is preserved under both spectral deformation and Baecklund transformation, for special choices of parameters, defining, in particular, transformations of constant mean curvature surfaces into new ones, with preservation of the mean curvature, in the latter case.