manifold $(X,J)$ is chosen, one defines the $\omega $-area of a $J$-complex curve $u:(S,J_S)\to (X,J)$ as $$area[u(S)] = \int_Su^*\omega .\eqno(1.3.4)$$ Remark that if $\omega $ is symplectic, then by [*Corollary 1.3.1*]{} this area coincides with $g$-area for the metric $g(\cdot ,\cdot ) =
\omega (\cdot ,J\cdot )$ and $J$-complex curves are $g$-minimal surfaces.

Remark also that for a $J$-complex curve $u:(\Delta ,J\st )\to (X,J)$, parameterized by a standard unit disk, one has $$\int_{\Delta }u^*\omega = \int_{\Delta }\omega  (du({\d \over \d x}),
du({\d \over \d y}))dx\wedge dy =  -\int_{\Delta }{1\over 2}g(du({\d \over
\d x}),Jdu(i{\d \over \d x}))dx\wedge dy =$$ $$=-\int_{\Delta }{1\over 2}g(du({\d \over \d x}),J^2du({\d \over \d x}))dx
\wedge dy = \int_{\Delta }{1\over 2}g({\d u\over \d x}, {\d u \over \d x}))
dx\wedge dy = {1\over 2}\Vert {\d u\over \d x}\Vert^2_{L^2(\Delta ,X)}.$$ Thus $$\int_{\Delta }u^*\omega  = \Vert {\d u\over \d x}
\Vert^2_{L^2(\Delta ,X)}+\Vert {\d u\over \d y}\Vert^2_{L^2(\Delta ,X)}.
\eqno(1.3.5)$$ The right hand side of (1.3.5) is called an energy of a $C^0\cap L^{1,2}$-map (not necessary holomorphic!) from $\Delta $ to a Riemannian manifold $(X,g)$. Thus, (1.3.5) tells us that, for holomorphic maps, energy and the area of the image, taken with multiplicities, are equal.

Remark. Let $g$ be a Riemannian metric 