Hermitian harmonic maps from complete Hermitian manifolds to complete Riemannian manifolds
In Acta Math. 170, No. 2, 221-254 (1993; Zbl 0806.53064), \textit{J. Jost} and \textit{S.-T. Yau} introduced a nonlinear elliptic system (H) for maps from a Hermitian manifold \(M\) to a Riemannian manifold \(N\). The solutions of (H) are called Hermitian harmonic maps. When \(M\) is not Kähler, then holomorphic maps are not necessarily harmonic, but always Hermitian harmonic. Jost and Yau also gave an existence and uniqueness theorem for Hermitian harmonic maps homotopic to a given \(h: M\to N\) relative to \(\partial M\), when \(N\) is nonpositively curved. In the present paper, the results of Jost and Yau are generalized to the case where \(M\) is no longer compact but complete. Here, compactness of \(M\) is replaced by positivity of the first eigenvalue of the Laplacian associated to a fixed holomorphic torsion-free connection and certain \(L^p\) bounds on the given data. The author also gives conditions for the global existence of the Hermitian harmonic map heat flow associated to (H) and starting from \(h\). Under certain additional assumptions, this heat flow converges to a Hermitian harmonic map.
scientific article; zbMATH DE number 1384392