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

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.

💡 Research Summary

The paper investigates the regularity of solutions to Dirichlet problems for complex Monge‑Ampère equations on Stein spaces that possess isolated singularities. Let (X) be a reduced, locally irreducible complex analytic space of complex dimension (n\ge 1) with a single isolated singular point (X_{\rm sing}). A Hermitian metric with fundamental form (\beta) induces a distance (d_\beta). Inside (X) we consider a bounded, strongly pseudoconvex open set (\Omega) defined by a smooth strictly plurisubharmonic exhaustion function (\rho) ((\Omega={\rho<0})). The boundary data (\varphi) is continuous on (\partial\Omega) and the density (f) belongs to (L^{p}(\Omega,\beta^{n})) with (p>1). The Dirichlet problem is

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