1.3.1. [*Let $(X,w,J)$ be a symplectic manifold with $w$ being $J$-calibrated. Then $J$-complex submanifolds are minimal and their volume is given by $$vol_{2k}(Y) = {1\over k!}\int_Yw^k. \eqno(1.3.2)$$*]{}

Proof. Let $Y$ be a $J$-complex submanifold of $X$, and $Y_1$ some submanifold of dimension $2k$ as well as $Y$. Suppose $\d Y=\d Y_1$ and $Y\cup (-Y_1) \sim 0$. Denote by $dy$ and $dy_1$ the volume forms of $Y$ and $Y_1$ with respect to metric $g(u,v)=\omega (u,Jv)$. Then we have $$0=    \int_{Y\cup (-Y_1)}\sigma_k = \int_{Y}\sigma_k  - \int_{Y_1}\sigma_k  =
\int_{Y}\sigma_k (TY)dy - \int_{Y_1}\sigma_k(TY_1)dy_1\ge$$ $$\ge \int_{Y}dy  - \int_{Y_1}dy_1=vol(Y)-vol(Y_1),$$ with equality taking place if and only if $\sigma_k(T_pY_1)=1$ for all $p\in Y_1$, , when $Y_1$ is also $J$-complex.

Definition 1.3.2. [*A $C^0\cap L^{1,2}$-map $u :(Y,J_Y)
\to (X,J_X)$ is called holomorphic if for a.-a. $x\in Y$ $$du_x\scirc J_Y(x)= J_X(u(x))\scirc du_x\eqno(1.3.3)$$ as mappings $T_xY\to T_{u(x)}X$.* ]{}

Exercise. [Check that for complex valued functions, , $u :(\rr^2,i)\to (\rr^2,i)$, (1.3.3) is a Cauchy-Riemann equation. ]{}

Sometimes one calls such $u$ a $(J_Y,J_X)$-holomorphic map. In the special case when $(Y,J_Y)$ is a Riemann surface $(S,J_S)$, one calls $(S,J_S,u)$ a parameterized $J_X$-complex curve. Its image $u(S)$ one simply calls a $J_X$-complex curve.

If some $J$-calibrated exterior $2$-form $\omega $ on an almost-complex 