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

数学物理学报(英文版) ›› 2018, Vol. 38 ›› Issue (5): 1468-1484.

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

陈兵龙, 朱熹平

Department of Mathematics, Sun Yat-sen University, Guangzhou 510275, China

收稿日期:2018-01-19 出版日期:2018-11-09 发布日期:2018-11-09
作者简介:Binglong CHEN,E-mail:mcscbl@mail.sysu.edu.cn;Xiping ZHU,E-mail:stszxp@mail.sysu.edu.cn
基金资助:
This work was partially supported by NSFC (11521101, 11025107).

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

Binglong CHEN, Xiping ZHU

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).

摘要/Abstract

摘要： 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.

关键词: uniformization conjecture, non-maximal volume growth, Chern number

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

陈兵龙, 朱熹平. YAU'S UNIFORMIZATION CONJECTURE FOR MANIFOLDS WITH NON-MAXIMAL VOLUME GROWTH[J]. 数学物理学报(英文版), 2018, 38(5): 1468-1484.

Binglong CHEN, Xiping ZHU. YAU'S UNIFORMIZATION CONJECTURE FOR MANIFOLDS WITH NON-MAXIMAL VOLUME GROWTH[J]. Acta Mathematica Scientia, 2018, 38(5): 1468-1484.

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

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

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