2008/01/22 by Jianguo Cao, Cao, JIanguo, Shu-Cheng Chang +1
Mathematics · #53C20 #53C23 #Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.0801.3431
openalex publication_date 2008/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper, we derive a partial result related to a question of Yau: "Does a simply-connected complete Kähler manifold M with negative sectional curvature admit a bounded non-constant holomorphic function?" Main Theorem. Let M2n be a simply-connected complete Kähler manifold M with negative sectional curvature ≤ -1 and S_∞(M) be the sphere at infinity of M. Then there is an explicit \it bounded contact form β defined on the entire manifold M2n. Consequently, the sphere S_∞(M) at infinity of M admits a \it bounded contact structure and a bounded pseudo-Hermitian metric in the sense of Tanaka-Webster. We also discuss several open modified problems of Calabi and Yau for Alexandrov spaces and CR-manifolds.