The well-known Yau’s uniformization conjecture states that any
complete noncompact Kaehler manifold with positive bisectional curvature is
bi-holomorphic to the complex Euclidean space. The conjecture for the case
of maximal volume growth has been recently confirmed by G. Liu. In this
talk, we will consider the conjecture for manifolds with non-maximal volume
growth. We will show that the finiteness of the first Chern number is an
essential condition to solve Yau’s conjecture by using algebraic embedding
method. Furthermore, we can verify the finiteness in the case of minimal
volume growth. In particular, we obtain a partial answer to Yau’s
uniformization conjecture on complex two-dimensional Kaehler manifolds with
minimal volume growth. This is a joint work with Bing-Long Chen.