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 equation 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 for sufficiently large power the solution decays as quickly as linear waves. Key ingredients for the proof include vector field method, conformal compactification and the geometric bilinear trace theorem for null hypersurface developed by Klainerman-Rodnianski.