Let $(X,g_X)$ be a Riemann surface and $(Y,g_Y)$ a Kahler manifold. Let:

$\phi\colon X\to Y$

be a minimal immersion, that is, a conformal harmonic smooth map with respect to $g_X$ and $g_Y$. If I am not mistaken, every holomorphic immersion from a Riemann surface to a Kahler manifold is a minimal immersion. I am interested in the opposite question: I would like to know the weaker set of sufficient conditions currently available in the literature (such that compactness, curvature conditions etc) on $(X,g_X)$ or $(Y,g_Y)$ that guarantees that such minimal immersion is holomorphic. The literature on these beautiful topics is huge, so it is not so easy to dive in and cleanly extract a number of clear necessary conditions for the case I am interested in.

Thanks.