on $C$ compatible with $j_C$, $h$ a Riemannian metric on $X$, and $u:C
\to X$ a $J$-holomorphic immersion. Then $\norm{du}^2_{L^2(C)}$ is independent of the choice of $g$ and coincides with the area of the image $u(C)$ the metric $h_J(\cdot, \cdot) \deff \half(h(\cdot, \cdot) + h(J\cdot, J\cdot)
)$. The metric $h_J$ here can be seen as a “Hermitization” of $h$ $J$. The independence of $\norm{du}^2_{L^2(C)}$ of the choice of a metric $g$ on $C$ in the same conformal class is a well-known fact, see, , \[S-U\]. Thus we can use the flat metric $dx^2 + dy^2$ to compare the area and the energy. For a $J$-holomorphic map we get $$\norm{du}^2_{L^2(C)}= \int_C |\d_x u|_h^2 + |\d_y u|_h^2 =
\int_C |\d_x u|_h^2 + |J \d_x u|_h^2 = \int_C |du|_{h_J}^2 = 
\area_{h_J}(u(C)),$$ where the last equality is another well-known result, see, , \[G\]. Since we consider changing almost complex structures on $X$, it is useful to know that we can use any Riemannian metric on $X$ having a reasonable notion of area.

[1.4. Symplectic Surfaces.]{}

First we recall some elementary facts about orthogonal complex structures in $\rr^4$.

In $\rr^4$ with coordinates $x_1, y_1, x_2, y_2$ consider the standard symplectic form $\omega\st  = dx_1\wedge dy_1 