Singular Solutions to the Complex Monge-Ampère Equation

Abstract

We present an explicit pluripotential and viscosity solution to the complex Monge-Ampère equation with constant right-hand side on $\mathbb{D}\times {\mathbb{C}}^{n-1}(n\ge 2)$, which lies merely in $W_{loc}^{1,2}\cap W_{loc}^{2,1}$ and is not even Dini continuous. Additionally, we exhibit two families of explicit entire toric solutions on ${\mathbb{C}}^n$ with continuous Hölder exponent $\hat{I}\pm \in (0,1)$ and $W_{loc}^{2,p}$ exponent $p\in (1,2)$.

12 pages. v2: exposition improved, several typos corrected