Yau's uniformization conjecture for manifolds with non-maximal volume growth

The well-known Yau's uniformization conjecture states that any complete noncompact Kähler 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 $C_1^n$ is an essential condition to solve Yau's conjecture by using algebraic embedding method. Moreover, we prove that, under bounded curvature conditions, $C_1^n$ 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 Kähler manifolds with minimal volume growth.