发表机构
Department of Mathematics, University of Trento; Institute of Mathematics, University of Warsaw（特伦托大学数学系; 华沙大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明次黎曼李群间的调和态射是光滑共形淹没且为拉普拉斯算子的对称性，给出李群外调和态射的部分结果，计算其水平能量第一变分，发现其不一定是调和映射。
AI 中文摘要
我们证明，次黎曼李群之间的调和态射是光滑映射，是次黎曼拉普拉斯算子的对称性，也是满足特定偏微分方程的共形淹没。此外，我们给出了李群之外调和态射的一些部分结果：证明了调和齐性空间存在调和坐标，且所有次黎曼流形中的光滑调和态射都是拉普拉斯算子的对称性。最后，我们计算了次黎曼李群上调和态射的水平能量的第一变分：它由目标李代数上的线性形式（模失配）给出，该形式在黎曼目标和卡尔诺（Carnot）目标中消失，但一般情况下并不消失。因此，与黎曼情形不同，次黎曼李群的调和态射不一定是调和映射。
英文摘要
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.
