The Bonnet problem for harmonic maps to the three-sphere

Abstract

Abstract In previous work [15] the authors defined transforms for non-conformal harmonic maps from a Riemann surface into the 3-sphere. An observation from that study was that two invariants, a real and a complex function, determine a non-conformal harmonic map up to isometries of the 3-sphere. We now show that if the first invariant of a harmonic map and its transformed map are the same, then these maps are either congruent or the harmonic map belongs to a particular 1-parameter family. Inspired by this result we discuss the Bonnet problem for non-conformal harmonic maps: to what extent is a harmonic map determined by its first invariant?