On the structure of complete Kähler manifolds with nonnegative curvature near infinity
The main purpose of this article is to study the geometric structure at infinity and the topology of a complete noncompact Kähler manifold M whose sectional curvature is nonnegative outside a compact subset (or for the sake of convenience near infinity). More precisely, the fact is established that if M is of complex dimension m and has more than one end, then except on a compact set, M is metrically a product of a complete complex 1-manifold \(\Sigma\) with a compact Kähler manifold N. The manifold \(\Sigma\) has nonnegative curvature near infinity and hence must be conformally equivalent to a Riemann surface with finite punctures. The compact manifold N has nonnegative sectional curvature and hence its universal cover \(\tilde N\) must be isometrically a product of Kähler manifolds which are either i) isometrically \({\mathbb{C}}^ k\), ii) isometrically an irreducible compact Hermitian symmetric space of rank not less than 2, or iii) isometrically a \({\mathbb{C}}{\mathbb{P}}^ k\) with a nonnegatively curved Kähler metric. The main technique is to utilize the nonconstant positive harmonic functions. The cases when M has more than one end with one of it being a large end and if M has one large end are considered. The first homology group of M is studied and the first Betti number is estimated from above by 2m-1, where m is the complex dimension of M. In addition, if M has positive sectional curvature near infinity and has a large end, then its first homology group must be trivial.
scientific article; zbMATH DE number 4138717
scientific article