GREEN'S FUNCTIONS AND COMPLEX MONGE-AMPERE EQUATIONS

Uniform L-1 and lower bounds are obtained for the Green's function on compact Kahler manifolds. Unlike in the classic theorem of Cheng-Li for Riemannian manifolds, the lower bounds do not depend directly on the Ricci curvature, but only on integral bounds for the volume form and certain of its derivatives. In particular, a uniform lower bound for the Green's function on compact Kahler manifolds is obtained which depends only on a lower bound for the scalar curvature and on an L-q norm for the volume form for some q > 1. The proof relies on auxiliary Monge-Amp & egrave;re equations, and is fundamentally non-linear. The lower bounds for the Green's function imply in turn C-1 and C-2 estimates for complex Monge-Amp & egrave;re equations with a sharper dependence on the function on the right hand side.