A note on complex Monge-Ampère equation in Stein manifolds
This paper studies the complex Monge-Ampère equation in Stein manifolds with boundary. More precisely, the author shows that given a compact \(n\)-dimensional Stein manifold with boundary \(X\), a \((1,1)\)-Kähler form \(\omega\) on \(X\), a function \(f\in C^\infty(X)\), and a function \(\phi\in C^\infty(\partial X)\), there exists a unique smooth function \(u\in C^\infty(X)\) such that \(\omega+i\partial\overline{\partial}u>0\) solving the complex Monge-Ampère equation \((\omega+i\partial\overline{\partial}u)^n=e^f\omega^n\) and satisfying the boundary condition \(u| _{\partial X}=\phi\).
scientific article; zbMATH DE number 5312750