= Solution
Use the compatible metrics $g_\Sigma(\cdot,\cdot)=\omega_\Sigma(\cdot,j\cdot)$ and $g_X(\cdot,\cdot)=\omega_X(\cdot,J\cdot)$. The energy of a smooth map is
$$
E(u)=\frac12\int_\Sigma|du|^2\,\omega_\Sigma.
$$
It is a <J-holomorphic curve> when $du\circ j=J\circ du$. Splitting $du$ into its complex-linear and complex-antilinear parts gives the <Energy identity for a J-holomorphic curve>
$$
E(u)=\int_\Sigma u^*\omega_X+ ext{a nonnegative multiple of }\|\bar\partial_Ju\|_{L^2}^2.
$$
The first term depends only on the homology class. It follows that a J-holomorphic map minimizes energy among all maps in its homology class.
One <Monotonicity theorem for a J-holomorphic curve> says that a nonconstant J-holomorphic curve through the center of a sufficiently small radius-$\rho$ ball, with boundary outside that ball, has area at least $c\rho^2$; in the standard complex ball one may take the sharp value $\pi\rho^2$. The <Gromov non-squeezing theorem> says
$$
B^{2n}(R)\hookrightarrow B^2(r)\times\mathbb R^{2n-2}
\quad\Longrightarrow\quad R\leq r.
$$
Write $\mathbb R^4=T^*\mathbb R^2$ with coordinates $(q,p)$ and form $dq_1\wedge dp_1+dq_2\wedge dp_2$. For $c>0$, the graph
$$
L_c=\{(q,cq):q\in\mathbb R^2\}
$$
is a <Lagrangian subspace>. Points of $L_c\cap B(R)$ have $|q|\leq R/\sqrt{1+c^2}$. Choose $c$ so large that
$$
\frac{2R}{\sqrt{1+c^2}}+2\varepsilon<1.
$$
Then no two points of the $\varepsilon$-neighborhood of $L_c\cap B(R)$ differ by a nonzero vector $(m,0)$ with $m\in\mathbb Z^2$, so quotienting $q$ modulo $\mathbb Z^2$ is injective there.
Relative to the Lagrangian splitting $L_c\oplus JL_c$, the map $(v,w)\mapsto(\lambda v,\lambda^{-1}w)$ is symplectic. It sends $B^4(r)$ into the indicated long thin neighborhood whenever
$$
\lambda r<R,
\qquad
r/\lambda<\varepsilon.
$$
These inequalities are compatible when $R>r^2/\varepsilon$. Taking such an $R$ and then the quotient constructs the <arbitrarily large symplectic balls in a cotangent cylinder>:
$$
\boxed{B^4(r)\hookrightarrow T^2\times\mathbb R^2\text{ for every }r>0.}
$$
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