Viscosity solutions to degenerate complex monge-ampère equations

Abstract

Degenerate complex Monge-Ampère equations on compact Kähler manifolds have recently been studied intensively using tools from pluripotential theory. We develop an alternative approach based on the concept of viscosity solutions and systematically compare viscosity concepts with pluripotential theoretic ones. This approach works only for a rather restricted type of degenerate complex Monge-Ampère equations. Nevertheless, we prove that the local potentials of the singular Kähler-Einstein metrics previously constructed by the authors are continuous plurisubharmonic functions. They were previously known to be locally bounded. Another application is a lower-order construction with a C0-estimate of the solution to the Calabi conjecture that does not use Yau's celebrated theorem. © 2011 Wiley Periodicals, Inc.