Constant mean curvature tori as stationary solutions to the Davey-Stewartson equation

TL;DR

This paper establishes a connection between constant mean curvature tori and their stationary behavior under the Davey-Stewartson flow, extending classical results from curves to surfaces in conformal geometry.

Contribution

It proves that constant mean curvature tori are exactly the stationary solutions to the Davey-Stewartson equation in the conformal 3-sphere.

Findings

Constant mean curvature tori are stationary solutions to the Davey-Stewartson flow.

Extension of classical curve results to surface case in conformal geometry.

Characterization of surfaces via integrable flow stationary conditions.

Abstract

A well known result of Da Rios and Levi-Civita says that a closed planar curve is elastic if and only if it is stationary under the localized induction (or smoke ring) equation, where stationary means that the evolution under the localized induction equation is by rigid motions. We prove an analogous result for surfaces: an immersion of a torus into the conformal 3-sphere has constant mean curvature with respect to a space form subgeometry if and only if it is stationary under the Davey-Stewartson flow.