Asymptotic decay for the Chern-Simons-Higgs equations

In this paper, we study the long time asymptotic behaviors for solutions to the Chern-Simons-Higgs equations with a pure power defocusing nonlinearity. We obtain quantitative inverse polynomial time decay for the potential energy for all data with finite conformal energy. Consequently, the solution decays in time in the pointwise sense for all power. We also show that the solution decays as quickly as linear waves for sufficiently large powers. Key ingredients for the proof include the vector field method, conformal compactification, and the geometric bilinear trace theorem for null hypersurfaces developed by Klainerman and Rodnianski (2006).