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.