In this paper we consider the question of characterizing compact quotients of the complex 2-ball by curvature conditions. It was conjectured by Frankel [2] that the complex projective space can be characterized as a compact K/ihler manifold with positive holomorphic bisectional curvature. This conjecture was recently proved by Mori [4] and Siu-Yau [6]. The confirmation of the Frankel conjecture leads one to ask whether there is a corresponding curvature characterization for compact quotients of the complex ball which is the noncompact counterpart of the complex projective space. Mostow-Siu [5] constructed a compact Kfihler surface which has negative sectional curvature and yet whose universal covering is not biholomorphic to the complex 2-ball. Because of this example the Frankel conjecture does not have a strictly analogous noncompact counterpart. One has to impose additional conditions. The additional condition we consider in this paper is that the metric whose curvature is negative is Kfihler-Einstein. We will consider only the case of complex dimension two.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.