We will discuss the role of complex Monge-Ampere equations as auxiliary equations in deriving sharp analytic and geometric estimates in Kahler geometry. By studying Green's functions, we will explore how to derive estimates for diameters and establish uniform Sobolev inequalities on Kahler manifolds, which depend only on the entropy of the volume form and are independent of the lower bound of the Ricci curvature. This is based on a series of joint works with Phong, Song and Sturm.