Hermitian harmonic maps
Let \((X,\gamma)\) be a Hermitian manifold of complex dimension \(m\), equipped with the Hermitian metric \(\gamma\) and with local complex coordinates \((z^\alpha)_{\alpha= 1,\dots, m}\), and \((N,g)\) a Riemannian manifold of real dimension \(n\), with the Riemannian metric \(g\) and local real coordinates \((x^i)_{i= 1,\dots, n}\). Following \textit{J. Jost} and \textit{S. T. Yau} [Proc. Symp. Pure Math. 54, Part I, 245-280, Am. Math. Soc. (1993; Zbl 0806.58012)], a smooth map \(\phi: X\to N\) is said to be a Hermitian harmonic map if \(\text{trace}_\gamma(\nabla^N_{d\phi} d\phi)= 0\), where \(\nabla^N\) is the Lévi-Cività connection of \((N,g)\). As a parallel with (real) harmonic morphisms, in this paper the author introduces the notion of Hermitian harmonic morphisms as maps pulling local Hermitian harmonic functions back to Hermitian harmonic functions when \((N,g)\) is a Hermitian manifold, and characterizes them as Hermitian harmonic maps satisfying a geometrical condition called the biholomorphic horizontal weak conformality. The author then considers the case of maps mixing (classical) harmonicity and Hermitian harmonicity, i.e., pulling local harmonic functions back to Hermitian harmonic functions, and shows them to be precisely \(d''\)-horizontally weakly conformal Hermitian harmonic maps. Finally, the author illustrates the different notions discussed with a few examples.
biholomorphic horizontal weak conformality
scientific article; zbMATH DE number 1549683