N ov 2 00 2 Positively Curved Complete Noncompact Kähler Manifolds

In this paper we give a partial affirmative answer to a conjecture of Greene-Wu and Yau. We prove that a complete noncompact Kähler surface with positive and bounded sectional curvature and with finite analytic Chern number c 1 (M) 2 is biholomorphic to C 2. The celebrated theorem of Cheeger–Gromoll–Meyer [3], [10] states that a complete noncompact Riemannian manifold with positive sectional curvature is diffeomorphic to the Euclidean space. It is well-known that there is a vast variety of biholomorphically distinct complex structures on R 2n for n > 1 (see [2], [7]). To understand the relationship between the Riemannian structure and the complex structure on complete noncompact manifolds, we restrict attention to Kähler manifolds which has the effect of insuring a closer relationship between these two structures. In [8], Greene and Wu proved that a complete noncompact Kähler manifold with positive sectional curvature is Stein. This fact thus motivated the following conjecture: