Let $(M,g_M)$ be a compact Kähler manifold with negative bisectional curvature. Let $\alpha : (S,g_S) \to M$ be a continuous map from a compact Riemannian manifold $(S,g_S)$.
Is $\alpha$ homotopic to a harmonic map?
If $g_M$ has non-positive sectional curvature, then this follows from the well-known result of Eells--Sampson. My question concerns whether the sectional curvature can be replaced with the bisectional curvature.
I know that Jost-Yau and many others have studied harmonic maps in Kähler geometry, but I can't find a specific answer to the above question. Thanks in advance.
