On the existence of Hermitian-harmonic maps from complete Hermitian to complete Riemannian manifolds

On non-Kähler manifolds the notion of harmonic maps is modified to that of Hermitian harmonic maps in order to be compatible with the complex structure. The resulting semilinear elliptic system is not in divergence form.The case of noncompact complete preimage and target manifolds is considered. We give conditions for existence and uniqueness of Hermitian-harmonic maps and solutions of the corresponding parabolic system, which observe the non-divergence form of the underlying equations. Numerous examples illustrate the theoretical results and the fundamental difference to harmonic maps.Support from the research focus Globale Methoden in der komplexen Geometrie under the auspices of Deutsche Forschungsgemeinschaft is gratefully acknowledged.Acknowledgement We are grateful to Wolf von Wahl (University of Bayreuth) for his suggestion to investigate Hermitian-harmonic maps on noncompact manifolds.Dedicated to Prof. E. Heinz on the occasion of his 80^th birthday