Complex solutions of Monge-Amp\`ere equations

TL;DR

This paper introduces a method to transform 4-variable Monge-Ampère PDEs into complex PDEs in 2 variables, enabling explicit solutions for several important equations in differential geometry and mathematical physics.

Contribution

It presents a novel reduction technique for Monge-Ampère equations, allowing explicit construction of solutions for special Lagrangian, real Monge-Ampère, and Plebanski equations.

Findings

Explicit holomorphic solutions constructed

Reduction method demonstrated on key equations

Bridges real and complex PDEs in geometric analysis

Abstract

We describe a method to reduce partial differential equations of Monge-Amp\`ere type in 4 variables to complex partial differential equations in 2 variables. To illustrate this method, we construct explicit holomorphic solutions of the special lagrangian equation, the real Monge-Amp\`ere equations and the Plebanski equations.