Embedded Surfaces for Symplectic Circle Actions

The purpose of this article is to characterize symplectic and Hamiltonian circle actions on symplectic manifolds in terms of symplectic embeddings of Riemann surfaces. More precisely, it is shown that (1) if $(M,\omega)$ admits a Hamiltonian $S^1$-action, then there exists a two-sphere $S$ in $M$ with positive symplectic area satisfying $\langle c_1(M,\omega), S]\rangle > 0$, and (2) if the action is non-Hamiltonian, then there exists an $S^1$-invariant symplectic $2$-torus $T$ in $(M,\omega)$ such that $\langle c_1(M,\omega), T]\rangle = 0$. As applications, the authors give a very simple proof of the following well-known theorem which was proved by Atiyah-Bott, Lupton-Oprea, and Ono: Suppose that $(M,\omega)$ is a smooth closed symplectic manifold satisfying $c_1(M,\omega)=\lambda \cdot \omega]$ for some $\lambda \in \R$ and $G$ is a compact connected Lie group acting effectively on $M$ preserving $\omega$. Then (1) if $\lambda < 0$, then $G$ must be trivial, (2) if $\lambda=0$, then the $G$-action is non-Hamiltonian, and (3) if $\lambda > 0$, then the $G$-action is Hamiltonian.