Singular Solutions to the Complex Monge-Amp\`ere Equation

TL;DR

This paper constructs explicit singular solutions to the complex Monge-Ampère equation, including pluripotential, viscosity, and toric solutions with specific regularity properties, advancing understanding of solutions with minimal regularity.

Contribution

It provides explicit examples of singular solutions to the complex Monge-Ampère equation with low regularity, including new pluripotential, viscosity, and toric solutions.

Findings

Constructed a pluripotential and viscosity solution with minimal regularity.

Presented two families of explicit entire toric solutions with specific Hölder and Sobolev regularities.

Abstract

We present an explicit pluripotential and viscosity solution to the complex Monge-Amp\`ere equation with constant right-hand side on $\mathbb D\times\mathbb C^{n-1}\,(n\geq 2)$, which lies merely in $W^{1,2}_{loc}\cap W^{2,1}_{loc}$ and is not even Dini continuous. Additionally, we exhibit two families of explicit entire toric solutions on $\mathbb C^n$ with continuous H\"older exponent $\alpha\in(0,1)$ and $W^{2,p}_{loc}$ exponent $p\in (1,2)$.