PDEs from matrices with orthogonal columns

TL;DR

This paper investigates a system of third order PDEs related to convex functions, linking geometric structures like Hessian metrics and symmetric spaces to matrices with orthogonal columns, with applications in Poisson geometry and Kahler structures.

Contribution

It introduces a geometric framework for analyzing third order PDEs using Hessian metrics and symmetric spaces, connecting PDE solutions to matrix properties and geometric structures.

Findings

Explicit solutions for the PDE system

Characterization of generic and non-generic solutions

Applications to Poisson geometry and Kahler structures

Abstract

We discuss a system of third order PDEs for strictly convex smooth functions on domains of Euclidean space. We argue that it may be understood as a closure of sorts of the first order prolongation of a family of second order PDEs. We describe explicitly its real analytic solutions and all the solutions which satisfy a genericity condition; we also describe a family of non-generic solutions which has an application to Poisson geometry and Kahler structures on toric varieties. Our methods are geometric: we use the theory of Hessian metrics and symmetric spaces to link the analysis of the system of PDEs with properties of the manifold of matrices with orthogonal columns.