The Uniformization Problem for Complex Manifolds

The classical uniformization theorem of Riemann surfaces says that a simply connected Riemann surface is conformally equivalent to the Riemann sphere, the complex line, or the open unit 1-disc. As is well-known, this beautiful fact fails in higher dimensions. The set of complex structures on a nice contractible higher-dimensional complex manifold is huge and cannot be classified in such a simple manner. Therefore, in order to find a simple analog of the above uniformization theorem, one must impose more restrictions on the manifold. In this regard, one notices that the uniformization theorem can also be understood in differential geometry in the following manner. (i) Every compact surface with positive curvature is conformally equivalent to the Riemann sphere. (ii) Every noncompact complete surface with positive curvature is conformally equivalent to the complex line. (iii) Every simply connected complete surface with curvature bounded from above by a negative constant is conformally equivalent to the open unit 1-disc.