A study on Hamiltonian circle actions

Abstract

We present two results on symplectic manifolds with a Hamiltonian circle action. The first one is on the computation of the Gromov width. Let $(M, \omega)$ be a closed monotone symplectic manifold. Suppose there is a semifree Hamiltonian circle action on $(M, \omega)$ with isolated maximum. We prove that the Gromov width of $(M, \omega)$ is given by the difference of the maximum and the second maximum critical values of the moment map.

The second one is on the fixed point set of the action. Consider a 6-dimensional closed symplectic manifold with a semifree Hamiltonian circle action. If all fixed components are 2-dimensional, then the number of fixed surfaces of positive genus is 0, 1, 3, or 4.