Harmonic morphisms of sub-Riemannian Lie groups

Abstract: We prove that harmonic morphisms between sub-Riemannian Lie groups are smooth, are symmetries of the sub-Riemannian Laplacian and are conformal submersions that satisfy a particular PDE. Moreover, we give some partial results for harmonic morphisms beyond Lie groups. We show the existence of harmonic coordinates for harmonically homogeneous spaces, and that smooth harmonic morphisms are symmetries of the Laplacian in all sub-Riemannian manifolds. Finally, we compute the first variation of the horizontal energy along a harmonic morphism of sub-Riemannian Lie groups: it is given by a linear form on the Lie algebra of the target, the modular mismatch, which vanishes for Riemannian and Carnot targets, but surprisingly not in general. Therefore, unlike in the Riemannian case, harmonic morphisms of sub-Riemannian Lie groups need not be harmonic maps.