Quartic differentials and harmonic maps in conformal surface geometry

TL;DR

This paper explores the relationship between harmonic sphere congruences and special classes of surfaces in conformal geometry, characterizing when certain quartic differentials are divergence free and linking them to well-known surface types.

Contribution

It provides a new characterization of surfaces with divergence-free Bryant's quartic differential in terms of harmonic sphere congruences and classifies these surfaces within conformal surface geometry.

Findings

Harmonic sphere congruences relate to S-Willmore and quasi-umbilical surfaces.

Generically, Bryant's quartic differential is divergence free for superconformal or harmonic congruence orthogonal surfaces.

Characterization of surfaces with divergence-free quartic differential in conformal geometry.

Abstract

We consider codimension 2 sphere congruences in pseudo-conformal geometry that are harmonic with respect to the conformal structure of an orthogonal surface. We characterise the orthogonal surfaces of such congruences as either $S$-Willmore surfaces, quasi-umbilical surfaces, constant mean curvature surfaces in 3-dimensional space forms or surfaces of constant lightlike mean curvature in 3-dimensional lightcones. We then investigate Bryant's quartic differential in this context and show that generically this is divergence free if and only if the surface under consideration is either superconformal or orthogonal to a harmonic congruence of codimension 2 spheres. We may then apply the previous result to characterise surfaces with such a property.