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It seems to be a sort of mean value problem. There is a huge literature on that for harmonic maps on riemannian manifolds. Try to google "mean value theorem on manifolds" or something like that...
Anyway, since curvature in complex geometry decreases when passing to submanifolds, any surface which is a closed submanifold of a complex torus is an example of compact Kähler surface with non positive (whatever, except Riemannian sectional) curvature.
