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Let $\Sigma$ be a smooth open connected orientable real surface.

Are there obstruction results for the existence of an almost complex structure $J$ on $\Sigma$?

(Since almost complex structures are automatically integrable on surfaces, this is akin to asking for obstructions of an orientable open surface to be Riemann.)

Maybe I am missing something, but I am trying to get some clarity on the issue since the search results I've been able to find focus either on compact manifolds or higher-dimensional manifolds.

M.G.
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    Every surface admits a Riemannian metric, hence, a conformal structure, hence a holomorphic structure if it is oriented. This has nothing to do with compactness. I think this was discussed many times on MSE. – Moishe Kohan Apr 03 '24 at 16:13
  • @MoisheKohan: Ah, I see, thanks for setting me straight as I am not really familiar with conformal geometry. I guess this has to do with the fact that in the plane the conformal maps are precisely the biholomorphic maps (and anti-holomorphic maps are excluded b/c of orientability). Nevertheless, I'd be very thankful for a link to such a discussion if you happen to have one. – M.G. Apr 03 '24 at 16:26
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    Take a look: https://math.stackexchange.com/questions/1661331/proof-of-equivalence-of-conformal-and-complex-structures-on-a-riemann-surface/3569334#3569334 – Moishe Kohan Apr 03 '24 at 17:33
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    Also https://math.stackexchange.com/questions/3505821/what-does-it-mean-for-an-almost-complex-structure-to-be-compatible-with-a-rieman – Moishe Kohan Apr 03 '24 at 18:52
  • @MoisheKohan: thanks a lot for the links! The last observation that the Hodge star operator in this sort of degenerate case induces an almost complex structure (transferred via the musical isomorphisms) is especially slick! Anyhow, the question should now probably be closed as duplicate since, effectively, it has been already answered previously. – M.G. Apr 03 '24 at 21:56

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