Modulus of Continuity of Solutions to Complex Monge-Amp\`ere Equations on Stein Spaces

TL;DR

This paper investigates the regularity of solutions to complex Monge-Ampère equations on Stein spaces, showing they are Hölder continuous outside singularities when boundary data is Hölder continuous.

Contribution

It establishes Hölder continuity of solutions to complex Monge-Ampère equations on Stein spaces with isolated singularities, extending regularity results to singular complex spaces.

Findings

Solutions are Hölder continuous outside singular points.

Regularity depends on boundary data's Hölder continuity.

Results apply to equations with $L^p$ densities on Stein spaces.

Abstract

In this paper, we study the modulus of continuity of solutions to Dirichlet problems for complex Monge-Amp\`ere equations with $L^p$ densities on Stein spaces with isolated singularities. In particular, we prove such solutions are H\"older continuous outside singular points if the boundary data is H\"older continuous.