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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.

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    $\begingroup$ I know that you are interested in the other direction, but a holomorphic map from a Riemann surface to a Kähler manifold need not be an immersion. Maybe you meant to say that a holomorphic immersion from a Riemann surface to a Kähler manifold is a minimal immersion. Also, maybe you want a 'weaker set of sufficient conditions.... that guaranteee that a minimal immersion be holomorphic'. $\endgroup$
    – Robert Bryant
    Commented Apr 7, 2018 at 10:02
  • $\begingroup$ @RobertBryant Indeed you are right, my bad! Thanks! $\endgroup$
    – Bilateral
    Commented Apr 7, 2018 at 11:19
  • $\begingroup$ @RobertBryant About the necessity of assuming holomorphic immersion, that is what I thought first, but then I read page 8 of wrap.warwick.ac.uk/37004/1/WRAP_THESIS_Arezzo_1996.pdf which claims that every holomorphic map from a Riemann surface to a Kahler manifold is a minimal immersion. Perhaps in this particular case the "immersion" condition follows from holomorphicity? $\endgroup$
    – Bilateral
    Commented Apr 7, 2018 at 11:22
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    $\begingroup$ That claim is simply false, though; there are plenty of holomorphic maps from Riemann surfaces to Kähler manifolds that are not immersions. The author of the source you are reading was perhaps being careless. A correct statement is this: A non-constant holomorphic mapping from a connected Riemann surface to a Kähler manifold is an immersion outside of a discrete set of points in the domain. $\endgroup$
    – Robert Bryant
    Commented Apr 7, 2018 at 11:35
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    $\begingroup$ In some situations if the minimal surface is stable, then it is holomorphic. This paper summarize some cases in its introduction: numdam.org/item?id=ASNSP_2000_4_29_2_473_0 However, stability does not always implies that the map is holomorphic, as shown in sciencedirect.com/science/article/pii/… $\endgroup$
    – Vanderson Lima
    Commented Apr 22, 2018 at 19:58

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