The Modified Calabi-yau Problems for Cr-manifolds and Applications

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 M 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 bounded contact form β defined on the entire manifold M. Consequently, the sphere S∞(M) at infinity of M admits a 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.