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

作　　者：Binglong CHEN Xiping ZHU 陈兵龙;朱熹平(Department of Mathematics, Sun Yat-sen University)

机构地区：Department of Mathematics,Sun Yat-sen University,Guangzhou 510275,China

出　　处：《Acta Mathematica Scientia》2018年第5期1468-1484,共17页数学物理学报（B辑英文版）

基　　金：partially supported by NSFC(11521101,11025107)

摘　　要：The well-known Yau＇s uniformization conjecture states that any complete noncompact Kahler 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 thc 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 Kahler manifolds with minimal volume growth.The well-known Yau＇s uniformization conjecture states that any complete noncompact Kahler 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 thc 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 Kahler manifolds with minimal volume growth.

关 键 词：uniformization conjecture non-maximal volume growth Chern number 

分 类 号：O186.1[理学—数学]