Suppose I have a (let's say compact) symplectic manifold $(M, \omega)$ and I choose an $\omega$-tame almost-complex structure $J$ on $M$. (Edit: Actually, I think I only really care about the almost-complex structure $J$ here, and not at all about $\omega$.) My broad question is:

What is known about the existence theory of $J$-holomorphic curves in $M$?

This question is quite vague as stated. The reason for the ambiguity is that I have no idea what sort of theorems I should expect. Naively:

  • Thinking about $J$-holomorphic disks (and $J$-holomorphic curves with boundary more generally): I'd be interested in learning about (say) $J$-holomorphic versions of the Dirichlet Problem or the Plateau Problem.

  • Thinking about closed $J$-holomorphic spheres: I'd be interested in knowing when we can expect $J$-holomorphic curves to exist in every homotopy / homology class of $M$.

I'm guessing that at least some of these questions are completely naive -- true or false for obvious reasons, or completely hopeless, or just too ambiguous -- but I'd appreciate any clarification.

  • $\begingroup$ It's unclear to me what you are looking for here, or what you know already. The question of how many holomorphic curves there are in a given homology class (with constraints possibly) is given by Gromov-Witten invariants. The compactness theory of $J$-curves is nice mostly when you have the taming symplectic form. In partial answer to your second question, you know that holomorphic curves can only represent classes $A$ with $\omega(A) > 0$. $\endgroup$ – Sam Lisi Apr 30 at 0:17
  • $\begingroup$ Apologies for the vagueness; I'm asking several questions at once, I guess. I have essentially no knowledge of Gromov-Witten invariants, beyond the fact that they're a symplectic invariant given by some sort of count of $J$-curves. As I said in the post, I'm not very interested in the symplectic structure $\omega$, but rather in $J$ itself. Completely separately (I think), I don't know to what extent one can expect to solve the Plateau Problem for $J$-curves (e.g.: given a closed curve $\gamma \subset M$, is there a $J$-curve $f \colon \Sigma \to M$ with $\partial (f(\Sigma)) = \gamma$?). $\endgroup$ – Jesse Madnick Apr 30 at 0:54
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    $\begingroup$ The boundary value problem you have posed is not a good one. In general, the Cauchy-Riemann operator is Fredholm for totally real boundary condition. In other words, the boundary condition you want is $f(\partial \Sigma) \subset N \subset M$ where $N$ is a half-dimensional submanifold with the property that $JTN \oplus TN = TM$. Prescribing the curve $\gamma$ itself is over-determined. If you lose the symplectic structure, you are entering a very difficult territory -- the symplectic form is what is traditionally used to give the required a priori bounds. $\endgroup$ – Sam Lisi Apr 30 at 4:26
  • $\begingroup$ I think you should narrow the scope of your question to get something more well-defined. In the meantime, this survey article by Donaldson might be useful: ams.org/notices/200509/what-is.pdf If you are looking for something heftier, take a look at McDuff-Salamon. They have made an older version of a predecessor book available online: barnard.edu/sites/default/files/inline/jholsm_0.pdf $\endgroup$ – Sam Lisi Apr 30 at 4:39
  • $\begingroup$ @SamLisi: Thanks for your comments. I didn't appreciate the importance of the totally real boundary condition; that's very good to know. I have a copy of McDuff-Salamon -- which I've spent the last couple of months reading -- but haven't yet found the answers I'm looking for in there. Donaldson's article is interesting and enlightening, but also doesn't address my questions. $\endgroup$ – Jesse Madnick May 1 at 21:56

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