A Generalised Monge-Ampère Equation

Abstract

We consider a generalised complex Monge-Ampère equation on a compact Kähler manifold and treat it using the method of continuity. For complex surfaces we prove an existence result. We also prove that (for three-folds and a related real PDE in a ball in R^3) as long as the Hessian is bounded below by a pre-determined constant (whilst moving along themethod of continuity path), a smooth solution exists. Finally, we prove existence for another real PDE in a 3-ball, which is a local real version of a conjecture of X. X. Chen.