Positively Curved Complete Noncompact K\"{a}hler Manifolds
  In this paper we give a partial affirmative answer to a conjecture of Greene-Wu and Yau. We prove that a complete noncompact K\"ahler surface with positive and bounded sectional curvature and with finite analytic Chern number $c_{1}(M)^{2}$ is biholomorphic to ${\C}^2$. 
Bing-Long Chen, Xi-Ping Zhu