  In this paper, we prove that if the area functional of a surface $\Sigma^2$
in a symplectic manifold $(M^{2n},\bar{\omega})$ has a critical point or has a
compatible stable point in the same cohomology class, then it must be
$J$-holomorphic. Inspired by a classical result of Lawson-Simons, we show how
various restrictions of the stability assumption to variations of metrics in
the space "projectively induced" metrics are enough to give the desired
conclusion.
