Acta Mathematica Scientia ›› 2018, Vol. 38 ›› Issue (5): 1468-1484.

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

Binglong CHEN, Xiping ZHU   

  1. Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China
  • Received:2018-01-19 Online:2018-11-09 Published:2018-11-09
  • Supported by:
    This work was partially supported by NSFC (11521101, 11025107).

Abstract: 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 C1n is an essential condition to solve Yau's conjecture by using algebraic embedding method. Moreover, we prove that, under bounded curvature conditions, C1n 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.

Key words: uniformization conjecture, non-maximal volume growth, Chern number

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