Quartic differentials and harmonic maps in conformal surface geometry

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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 lightcone 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.
Original languageEnglish
Pages (from-to)1507
Number of pages1524
JournalAnnali della Scuola Normale Superiore di Pisa - Classe di Scienze
DOIs
Publication statusPublished - 30 Sep 2022

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