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.

中文翻译：

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谐波热带态射和近似

谐波变形虫是代数曲线在复杂圆环中的变形虫的推广。由 Krichever 在 2014 年引入，对这些物体的考虑建议扩大热带几何的范围。在本文中，我们介绍了从热带曲线到仿射空间的调和态射的概念，并展示了如何将这些态射系统地描述为黎曼曲面上调和变形虫映射族的极限。它扩展了先前关于仿射空间中热带曲线近似的结果，并提供了关于规则相热带态射的米哈尔金近似定理的不同观点，如米哈尔金在 2006 年所述。