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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.