The Asymptotic Behavior of Finite Energy Symplectic Vortices with Admissible Metrics

Assume $(X, \omega)$ is a compact symplectic manifold with a Hamiltonian compact Lie group action and the zero in the Lie algebra is a regular value of the moment map $\mu$. We prove that a finite energy symplectic vortex exponentially converges to (un)twisted sectors of the symplectic reduction at cylinder ends whose metrics grow up at least cylindrically fast, without assuming the group action on the level set $\mu^{-1}(0)$ is free. It generalizes the corresponding results by Ziltener [23, 24] under the free action assumption. The result of this paper is the first step in setting up the quotient morphism moduli space induced by the authors in [6]. Necessary preparations in understanding the structure of such moduli spaces are also introduced here. The quotient morphism constructed in [6] is a part of the project on the quantum Kirwan morphism by the authors (see [3, 4, 5]).