Modified energy functional characterizing the harmonic morphism equation

Construct a natural modification of the horizontal energy that incorporates the Haar measure of the target and whose critical points among contact maps are exactly the maps satisfying the harmonic morphism equation $(F)=0$.

Background

The paper shows that harmonic morphisms need not be critical points of the standard horizontal energy because that energy depends only on the target horizontal metric, whereas the harmonic morphism equation also depends on the target Haar measure through the modular mismatch. The authors identify the required correction to the first variation as −∫Ωλ2χ(φ)dvolG-\int_\Omega\lambda^2\chi(\varphi)dvol_G, but leave open whether it arises from a natural modified energy functional.

References

Is there a natural modification of the horizontal energy, involving the Haar measure of the target, whose critical points among contact maps are exactly the maps satisfying $(F)=0$?

— Harmonic morphisms of sub-Riemannian Lie groups  (2609.04299 - Golo et al., 3 Sep 2026) in Section 8, Subsection “Two questions,” Question \ref{hh:q:functional}