 Solution

No nonconstant J-holomorphic curve with this closed connected source exists. The standard symplectic form on ${\mathbb{R}}^{2n}$ is exact; for instanceThe Energy identity for a J-holomorphic curve and the Generalized Stokes theorem giveThe target metric from the compatible almost complex structure is positive definite, so the nonnegative continuous energy density must vanish everywhere. Thus $du=0$. Connectedness of $\Sigma$ makes $u$ constant. This argument uses exactness and compatibility, and works even when $J$ is nonintegrable; it does not require the ordinary holomorphic maximum principle on the target.