Let M be a 2n-dimensional closed symplectic manifold. Then is there a constant c such that , for any real 2-dimensional embedded J-holomorphic disk u, the symplectic volume of u is bounded by c ?
If not, is there any result about a condition which makes the statement above to be true? I really thank you for your any comment.
Unless I'm misunderstanding what you're asking, the answer is surely no...consider for instance $\mathbb{C}P^n$ with its standard symplectic and complex structures. This admits embedded $J$-holomorphic curves of arbitrarily large area (take a high-degree curve in a plane $\mathbb{C}P^2\subset \mathbb{C}P^n$), and restricting to a disc within any of these curves would give you a $J$-holomorphic disc $u$ of arbitrarily large area (which I assume is what you mean by the symplectic volume of $u$).
On more general symplectic manifolds $(M,\omega)$, the h-principle gives you immersed symplectic spheres in every homology class $A$ with $\int_{A}\omega>0$; these spheres can be taken embedded if $\dim M\geq 6$ and embedded away from finitely many transverse double points if $\dim M=4$. In either case you could construct an almost complex structure $J$ on $M$ with respect to which an arbitrarily large-area subdisk of the surface is embedded and $J$-holomorphic. (This is admittedly a little weaker than the first example, since here we're choosing $J$ after we choose the surface--so all it shows is that for any $C$ there is a pair $(u,J)$ where $u$ is a $J$-holomorphic disc of area larger than $C$, with $J$ possibly depending on $C$.)
Thank you for your kind answer. Actually, I study about a closed non-hamiltonian symplectic $S^1$-manifold $(M,\omega,J)$ with non-empty fixed point set.
I have one another question. Let X be a fundamental vector field of the action and let $\gamma(t) : [0, \infty) \to M$ be an integral curve for $JX$ with $\gamma(0) = z$, where $z$ is a fixed point for the given action. I wonder whether $\gamma(t)$ has the end point or not. (I mean whether $JX$-flows converges to another fixed point or not)
* If $\gamma(t)$ has no end point, then $S^1 \cdot \gamma([0,t])$ is a $J$-holomorphic disk for any $t \in (0,\infty)$. Hence I think it should have infinte symplectic volume. But as you said, this question seems not to be related to the symplectic volume of disk.
* In Hamiltonian action case, the integral curve $\gamma(t)$ of course converges to some fixed point by compactness of $M$
