A characterization of complex quasi-projective manifolds uniformized by unit balls

In 1988 Simpson extended the Donaldson–Uhlenbeck–Yau theorem to context of Higgs bundles, and as an application he proved a uniformization which characterizes complex projective manifolds quasi-projective curves whose universal coverings are unit balls. this paper we give necessary sufficient condition for be uniformized by This generalizes Simpson. Several byproducts also obtained in paper.