Monge-Ampère Equation

The Monge-Ampère equation is a type of mathematical formula used to describe how a shape, surface, or space can be transformed or mapped in a way that preserves specific volume or density properties. It involves a function whose second derivatives (curvature) relate to how the volume changes under the transformation. This equation appears in geometry, physics, and optimal transport, helping to understand how to optimally move or shape objects while maintaining certain constraints. Essentially, it captures the relationship between a surface's curvature and the distribution of volume in a precise, mathematically rigorous way.