Jacobi fields along harmonic maps

TL;DR

This paper investigates how Jacobi fields along harmonic maps preserve various geometric properties to first order, providing tools for understanding their integrability, especially in the context of maps from the 2-sphere to the complex projective plane.

Contribution

It demonstrates that Jacobi fields preserve conformality, holomorphicity, and isotropy properties to first order, advancing the understanding of their integrability in specific harmonic map settings.

Findings

Jacobi fields preserve conformality to first order

Jacobi fields preserve holomorphicity to first order

Jacobi fields preserve isotropy properties to first order

Abstract

We show that Jacobi fields along harmonic maps between suitable spaces preserve conformality, holomorphicity, real isotropy and complex isotropy to first order; this last being one of the key tools in the proof by Lemaire and the author of integrability of Jacobi fields along harmonic maps from the 2-sphere to the complex projective plane.