  We extend Eardley and Moncrief's $L^\infty$ 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 $L^\infty$ 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.
