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. 



symplectic volume of embedded J-holomorphic disk
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 $C$.)


