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

TL;DR

This paper establishes that critical or stable points of the area functional for surfaces in symplectic manifolds are necessarily $J$-holomorphic, linking geometric stability to complex structure compatibility.

Contribution

It provides a variational characterization of $J$-holomorphic curves using the area functional and stability conditions, extending classical results to symplectic settings.

Findings

Critical points of the area functional are $J$-holomorphic.

Stable points under certain metric variations are $J$-holomorphic.

The approach generalizes Lawson-Simons classical results.

Abstract

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.