Topological multi-vortex solutions of the Maxwell–Chern–Simons–Higgs model with a background metric

In this paper, we consider the self-dual equations arising from the Maxwell–Chern–Simons–Higgs model in a curved space with a background metric (1,−b(x),−b(x)). We assume that b(x) is not a constant and decays like |x|−γ with γ∈(0,2). Then, we prove that there exists a positive constant β∗ such that we have a topological solution of the self-dual equations if the couplings constants κ and q satisfy κq>β∗. We also verify the Chern–Simons limit which means that our solutions converge to the solution of the self-dual Chern–Simons vortex equation as q→∞.