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

Abstract

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

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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