Formal conserved quantities for isothermic surfaces

TL;DR

This paper introduces a formal Laurent series approach to conserved quantities of isothermic surfaces in spheres, providing new conformally invariant conditions for classifying special isothermic surfaces.

Contribution

It proves that all isothermic surfaces possess a formal Laurent series of parallel sections, extending the understanding of their conserved quantities and invariants.

Findings

Existence of a formal Laurent series of parallel sections for all isothermic surfaces.

Conformally invariant criteria for identifying special isothermic surfaces in $S^3$.

Extension of classical conservation laws to a formal series framework.

Abstract

Isothermic surfaces in $S^n$ are characterised by the existence of a pencil $\nabla^t$ of flat connections. Such a surface is special of type $d$ if there is a family $p(t)$ of $\nabla^t$-parallel sections whose dependence on the spectral parameter $t$ is polynomial of degree $d$. We prove that any isothermic surface admits a family of $\nabla^t$-parallel sections which is a formal Laurent series in $t$. As an application, we give conformally invariant conditions for an isothermic surface in $S^3$ to be special.