An ansatz for constructing explicit solutions of Hessian equations

TL;DR

This paper introduces a new ansatz that simplifies complex and real Hessian equations to ODE systems, enabling explicit solutions including entire and singular solutions for equations like dHYM, LYZ, Monge--Ampère, and special Lagrangian equations.

Contribution

The paper presents a novel ansatz reducing broad classes of Hessian equations to solvable ODE systems, yielding explicit solutions and detailed classifications.

Findings

Constructed explicit entire solutions for complex Hessian equations.

Produced singular solutions that extend to known special Lagrangian submanifolds.

Provided a unified approach to solving multiple Hessian equations explicitly.

Abstract

We introduce a (variation of quadrics) ansatz for constructing explicit, real-valued solutions to broad classes of complex Hessian equations on domains in $\mathbb{C}^{n+1}$ and real Hessian equations on domains in $\mathbb{R}^{n+1}$. In the complex setting, our method simultaneously addresses the deformed Hermitian--Yang--Mills/Leung--Yau--Zaslow (dHYM/LYZ) equation, the Monge--Amp\`{e}re equation, and the $J$-equation. Under this ansatz each PDE reduces to a second-order system of ordinary differential equations admitting explicit first integrals. These ODE systems integrate in closed form via abelian integrals, producing wide families of explicit solutions together with a detailed description. In particular, on $\mathbb{C}^{n+1}$, we construct entire dHYM/LYZ solutions of arbitrary subcritical phase, and on $\mathbb{R}^{n+1}$ we produce entire special Lagrangian solutions of arbitrary subcritical phase. Some of these solutions develop singularities on compact regions. In the special Lagrangian case we show that, after a natural extension across the singular locus, these blow-up solutions coincide with previously known complete special Lagrangian submanifolds obtained via a different ansatz.