A variational characterization of $J$-holomorphic curves in symplectic manifolds

In this paper, we prove that if the area functional of a surface ${\mathrm{\Sigma }}^2$ 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.