An ansatz for constructing explicit solutions of Hessian equations

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.