## 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.

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 language | English |
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Pages (from-to) | 1507 |

Number of pages | 1524 |

Journal | Annali della Scuola Normale Superiore di Pisa - Classe di Scienze |

DOIs | |

Publication status | Published - 30 Sep 2022 |