Modulus of Continuity of Solutions to Complex Monge-Ampère Equations on Stein Spaces

TL;DR

This work analyzes the modulus of continuity for solutions to complex Monge-Ampère Dirichlet problems on Stein spaces with isolated singularities, focusing on $L^p$ densities. The authors develop barrier methods and a resolution-based regularization scheme to control oscillations, combining barrier data with $L^1$-Laplacian-type estimates to obtain quantitative modulus bounds. The main result shows that, for $\phi\in C^0(\partial\Omega)$ and $f\in L^p(\Omega,\beta^n)$ with $p>1$, the unique solution $u$ satisfies $\omega_{u,x}(t) \le C_x \max\{ \omega_{\phi}(t^{1/2}), t^{1/(nq+1)}\}$, and if $\phi\in C^{0,\alpha}$, then $u$ is $\alpha^*$-Hölder outside the singular point with $\alpha^*<\min\{\alpha/2,1/(nq+1)\}$. This extends regularity theory for complex Monge-Ampère equations to mildly singular varieties and provides precise boundary and singularity-sensitive continuity behavior, with potential geometric applications in Kähler metrics on singular spaces.

Abstract

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

Theorems & Definitions (30)

• Definition 1.1
• Theorem 1.2
• Corollary 1.3
• Definition 1.4
• Proposition 1.5: Comparison principle
• Theorem 1.6
• Theorem 1.7
• Lemma 1.8
• Definition 2.1
• Lemma 2.2