Harmonic tropical morphisms and approximation

Harmonic amoebas are generalisations of amoebas of algebraic curves immersed in complex tori. Introduced by Krichever in 2014, the consideration of such objects suggests to enlarge the scope of tropical geometry. In the present paper, we introduce the notion of harmonic morphisms from tropical curves to affine spaces and show how these morphisms can be systematically described as limits of families of harmonic amoeba maps on Riemann surfaces. It extends previous results about approximation of tropical curves in affine spaces and provides a different point of view on Mikhalkin’s approximation Theorem for regular phase-tropical morphisms, as stated e.g. by Mikhalkin in 2006. The results presented here follow from the study of imaginary normalised differentials on families of punctured Riemann surfaces and suggest interesting connections with compactifications of moduli spaces.