Large data decay of Yang-Mills-Higgs fields on Minkowski and de Sitter spacetimes

We extend Eardley and Moncrief’s $L^{\mathrm{\infty }}$ estimates for the conformally invariant Yang-Mills-Higgs equations to the Einstein cylinder. Our method is to first work on Minkowski space and localise their estimates, and then carry them to the Einstein cylinder by a conformal transformation. By patching local estimates together, we deduce global $L^{\mathrm{\infty }}$ estimates on the cylinder, and extend Choquet-Bruhat and Christodoulou’s small data well-posedness result to large data. Finally, by employing another conformal transformation, we deduce exponential decay rates for Yang-Mills-Higgs fields on de Sitter space, and inverse polynomial decay rates on Minkowski space.