Continuous deformations of harmonic maps and their unitons

TL;DR

This paper investigates how harmonic maps into unitary groups can be smoothly deformed into maps with special symmetry properties, analyzing the analyticity and smoothness of the deformation process.

Contribution

It provides a detailed study of the deformation of harmonic maps into U(n), showing that the associated unitons vary smoothly and analytically with the deformation parameter.

Findings

Unitons depend smoothly on the deformation parameter.

The deformation is real analytic along lines through the origin.

Harmonic maps can be continuously deformed into S^1-invariant solutions.

Abstract

We consider harmonic maps on simply connected Riemann surfaces into the group $\mathrm{U}(n)$ of unitary matrices of order $n$. It is known that a harmonic map with an associated algebraic extended solution can be deformed into a new harmonic map that has an $S^1$-invariant associated extended solution. We study this deformation in detail and show that the corresponding unitons are smooth functions of the deformation parameter and real analytic along any line through the origin.