Boundedness of Differential of Symplectic Vortices in Open Cylinder Model

Let G be a compact connected Lie group, (X,ω,μ) a Hamiltonian G-manifold with moment map μ, and Z a codimension-2 Hamiltonian G-submanifold of X. We study the boundedness of the differential of symplectic vortices (A,u) near Z, where A is a connection 1-form of a principal G-bundle P over a punctured Riemann surface Σ˚, and u is a G-equivariant map from P to an open cylinder model near Z. We show that if the total energy of a family of symplectic vortices on Σ˚≅[0,+∞)×S1 is finite, then the A-twisted differential dAu(r,θ) is uniformly bounded for all (r,θ)∈[0,+∞)×S1.