YAU'S UNIFORMIZATION CONJECTURE FOR MANIFOLDS WITH NON-MAXIMAL VOLUME GROWTH

The well-known Yau's uniformization conjecture states that any complete non-compact K(a)hler manifold with positive bisectional curvature is bi-holomorphic to the Euclidean space.The conjecture for the case of maximal volume growth has been recently confirmed by G.Liu in [23].In the first part,we will give a survey on the progress.In the second part,we will consider Yau's conjecture for manifolds with non-maximal volume growth.We will show that the finiteness of the first Chern number Cn1 is an essential condition to solve Yau's conjecture by using algebraic embedding method.Moreover,we prove that,under bounded curvature conditions,Cn1 is automatically finite provided that there exists a positive line bundle with finite Chern number.In particular,we obtain a partial answer to Yau's uniformization conjecture on K(a)hler manifolds with minimal volume growth.