Timeline for When is a minimal immersion holomorphic?
Current License: CC BY-SA 3.0
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| Apr 22, 2018 at 19:58 | comment | added | Vanderson Lima | 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/… | |
| Apr 19, 2018 at 20:40 | comment | added | Pierre | I agree that this is not really an answer, as in what I describe the domain has to be high-dimensional... | |
| Apr 19, 2018 at 20:37 | comment | added | Pierre | Let me answer another question or rather give instances where a harmonic map between two Kähler manifolds is automatically holomorphic. If a map from a closed Kähler manifold to a complex hyperbolic manifold is harmonic and has real rank >2 at a point it is holomorphic (due to Carlson and Toledo). Here complex hyperbolic manifold = quotient of complex hyperbolic space. There is a similar statement due to Siu, changing complex hyperbolic space by another hermitian symmetric space of noncompact type (but "2" has to be replaced by a bigger constant then). | |
| Apr 7, 2018 at 11:35 | comment | added | Robert Bryant | 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. | |
| Apr 7, 2018 at 11:23 | history | edited | Bilateral | CC BY-SA 3.0 |
added 7 characters in body
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| Apr 7, 2018 at 11:22 | comment | added | Bilateral | @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? | |
| Apr 7, 2018 at 11:19 | comment | added | Bilateral | @RobertBryant Indeed you are right, my bad! Thanks! | |
| Apr 7, 2018 at 10:02 | comment | added | Robert Bryant | 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'. | |
| Apr 6, 2018 at 16:12 | history | asked | Bilateral | CC BY-SA 3.0 |