= Solution
\b[No nonconstant <J-holomorphic curve> with this closed connected source exists.] The <standard symplectic form> on $\mathbb R^{2n}$ is exact; for instance
$$
\omega_0=d\lambda_0,\qquad\lambda_0=\frac12\sum_j(x_jdy_j-y_jdx_j).
$$
The <Energy identity for a J-holomorphic curve> and the <Generalized Stokes theorem> give
$$
E(u)=\int_\Sigma u^*\omega_0
=\int_\Sigma d(u^*\lambda_0)
=\int_{\partial\Sigma}u^*\lambda_0=0.
$$
The 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.
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