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

Algebraic Geometry;Complex Variables