A fourth order family of minimal surfaces in the 3-sphere

TL;DR

This paper introduces a new family of minimal surfaces in the 3-sphere characterized by a fourth order equation, with geometric properties linked to surface of revolution metrics and flat 3-web structures.

Contribution

It defines a novel family of minimal surfaces in the 3-sphere via a compatible fourth order equation and analyzes their geometric properties and structure equations.

Findings

Surfaces are characterized by revolution-like metrics or flat 3-webs.

Structure equations decouple for a natural frame choice.

Analysis reduces to curves in the 2-sphere governed by a third order ODE.

Abstract

This is a preliminary note on a family of minimal surfaces in the 3-sphere defined by a compatible fourth order equation. The minimal surfaces are geometrically characterized either by having a surface of revolution like induced metric, or by having a flat structure 3-web. We observe that the structure equation un-couples for a natural choice of frame. The analysis is reduced to the associated curves in the 2-sphere defined by a rational third order ODE on the curvature.