On $L^{\infty }$ estimates for complex Monge-Ampère equations

Abstract

A PDE proof is provided for the sharp $L^{\infty }$ estimates for the complex Monge-Ampère equation which had required pluripotential theory before. The proof covers both cases of fixed background as well as degenerating background metrics. It extends to more general fully non-linear equations satisfying a structural condition, and it also gives estimates of Trudinger type.

24 pages