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

Claudio Arezzo; Jun Sun

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arXiv · 1211.7016

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

Claudio Arezzo·Jun Sun

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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.

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Claudio Arezzo, Jun Sun. 2012-11-29. A variational characterization of $J$-holomorphic curves in symplectic manifolds. https://arxiv.org/abs/1211.7016

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