Kähler-Einstein metrics and compactifications of Cn

The study of holomorphic and isometric immersions of a Kähler manifold (M,g) into a Kähler manifold (N,G) started with Calabi. In his famous paper [3] he considered the case when the ambient space (N,G) is a complex space form, i.e. its holomorphic sectional curvature KN is constant. There are three types of complex space forms : flat, hyperbolic or elliptic according as the holomorphic sectional curvature is zero, negative, or positive. Every n-dimensional complex space form is, after multiplying by a suitable constant, locally holomorphically isometric to one of the following: the complex euclidean space C with the flat metric; the unit ball in C with its Bergman metric of negative holomorphic sectional curvature; the ndimensional complex projective space CP endowed with the Fubini-Study metric gFS of positive holomorphic sectional curvature. Recall that gFS is the metric whose associated Kähler form is given by