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 K\"ahler manifold M with negative sectional curvature admit a bounded non-constant holomorphic function?"   Main Theorem. Let $M^{2n}$ be a simply-connected complete K\"ahler manifold M with negative sectional curvature $ \le -1 $ and $S_\infty(M)$ be the sphere at infinity of $M$. Then there is an explicit {\it bounded} contact form $\beta$ defined on the entire manifold $M^{2n}$.   Consequently, the sphere $S_\infty(M)$ 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. 
JIanguo Cao and Shu-Cheng Chang