Continuity of Monge–Ampère Potentials with Prescribed Singularities

Abstract

We study the continuity of solutions to complex Monge–Ampère equations with prescribed singularity type. This generalizes the previous results of Di Nezza–Lu and the author Dang (Int Math Res Not 14:11180-11201, 2022), Di Nezza and Lu (J Reine Angew Math 727:145-167, 2017). As an application, we can run the pluripotential Monge–Ampère flows in Dang (J Funct Anal 282(6):65, 2022) starting at a current with prescribed singularities.