Viscosity solutions to degenerate Complex Monge-Amp\`ere equations

TL;DR

This paper introduces a viscosity solutions framework for degenerate complex Monge-Ampère equations on compact Kähler manifolds, bridging viscosity and pluripotential theories, and proves continuity of solutions in the Kähler setting.

Contribution

It develops a new viscosity solutions approach for these equations and generalizes a continuity theorem from the projective to the Kähler case.

Findings

Viscosity solutions provide an alternative framework for degenerate complex Monge-Ampère equations.

Established a systematic comparison between viscosity and pluripotential solutions.

Proved continuity of solutions in the Kähler case, extending previous results.

Abstract

We develop an alternative approach to Degenerate complex Monge-Amp\`ere equations on compact K\"ahler manifolds based on the concept of viscosity solutions and compare systematically viscosity concepts with pluripotential theoretic ones. We generalize to the K\"ahler case a theorem due to Dinew and Zhang in the projective case to the effect that their pluripotential solutions constructed previously by the authors are continuous.