In this lecture, we discuss an open conjecture of Yau about the existence of non-trivial bounded holomorphic functions on simply-connected Kaehler manifolds with negative sectional curvature. Among other things, we apply Kohn's d-bar-b analysis on metric spheres with new uniform Holder estimates of CR functions developed by Shaw-Wang. In addition, we also use comparison theory of metric geometry t o construct bounded strictly pluri-subharmonic functions on 1-connected negatively Kaehler manifolds. The combination of comparison geometry and Kohn's CR-analysis on curved spaces will lead a new partial answer to this open problem of Yau, when complex dimension greater than or equal to 3.