  The problem involved in this paper is the global existence of the solution to
the $\mathfrak{su}(2)$-Einstein-Yang-Mills-Higgs(EYMH) equation. The approach
we employ stems from H. Lindblad and I. Rodnianski and is dependent of wave
coordinates and Lorentzian gauge conditions. Our main conclusion is that the
EYMH system admits global existence provided the initial datum are sufficiently
small. To the best of our knowledge, there is no similar result in the area of
EYMH equations.
