I have posted it on Mathstackexchange but nobody replied. 


Consider a loop $\gamma:\mathbb{S}^1\to M^{2n}$ in a symplectic manifold $(M^{2n},\omega)$. Let $J$ be an $\omega$-compatible almost complex structure on $M$.
My naive question is: when does $\gamma$ bound a pseudoholomorphic curve? More precisely,



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> when there does exists a Riemann surface with boundary $\Sigma$, and a J-holomorphic map $u:\Sigma\to (M,J)$ such that $u|\_{\partial \Sigma} \equiv \gamma$ ?
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I do not know much about pseudoholomorhic curves, I see that some people requires some conditions on the boundary like, contact type or totally real, if you want you can consider these assumptions or deal with these cases. 



Which curves are boundary of pseudoholomorphic curves?
The 'moment conditions' that Ben McKay mentions are simply this: A closed curve $C$ in $\mathbb{C}^n$ bounds a compact Riemann surface (which might be singular) if and only if the integral around $C$ of any global holomorphic $1$-form on $\mathbb{C}^n$ vanishes.


One direction is just Stokes' Theorem: If $\omega$ is a holomorphic $1$-form, then $\mathrm{d}\omega$ is a holomorphic $2$-form, and a holomorphic $2$-form vanishes when pulled back to any (possibly singular) complex curve. Thus, if $C = \partial X$ where $X\subset\mathbb{C}^n$ is a compact Riemann surface (with boundary), then $\int\_C\omega = \int\_X\mathrm{d}\omega = 0$.


The converse, which (I think) is due to Harvey and Lawson (*Boundaries of complex analytic varieties, I.*, Annals of Mathematics **102** (1975), 223–290), is rather deeper. 


I don't remember whether there is a test for when $C$ bounds an actual holomorphic disk (i.e., a Riemann surface of genus $0$). I believe I remember that when $C$ does bound a compact (singular) holomorphic Riemann surface, it bounds only one. (I don't have access to the Harvey-Lawson paper I cited just now. If you want the definitive statment, I suggest that you check that paper.)


