On the Local Regularity of Homogeneous Complex Monge-Ampère Equations

We establish a local higher regularity theorem for the homogeneous complex Monge-Ampere equation on $X\times D$, where $X$ is a complete, possibly noncompact Kahler manifold and $D\subset\mathbb C$ is the unit disc. For a bounded weak solution given by the Perron envelope, we prove higher regularity