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An ansatz for constructing explicit solutions of Hessian equations

Chung-Jun Tsai, Mao-Pei Tsui, Mu-Tao Wang

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--Ampère 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.

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 and real Hessian equations on domains in . In the complex setting, our method simultaneously addresses the deformed Hermitian--Yang--Mills/Leung--Yau--Zaslow (dHYM/LYZ) equation, the Monge--Ampère equation, and the -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 , we construct entire dHYM/LYZ solutions of arbitrary subcritical phase, and on 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.

Paper Structure

This paper contains 14 sections, 17 theorems, 141 equations.

Table of Contents

  1. Introduction
  2. The deformed Hermitian--Yang--Mills/Leung--Yau--Zaslow Equation
  3. First integrals of the ODE system
  4. Limits of $s$ when $n\geq 3$
  5. The isotropic Case
  6. Entire solutions of dHYM/LYZ
  7. When $\kappa$ is positive
  8. When $\kappa$ is negative
  9. On $\mathbb{C}^{n+1}$ with ${n+1\geq 4}$
  10. Solutions to the special Lagrangian equation
  11. Extensions of solutions to the special Lagrangian equation
  12. General equations of recursive type
  13. Complete classification when $n=3$
  14. Some Algebraic Calculations

Key Result

Theorem 1.2

Suppose complex H_eqn/real H_eqn is of recursive type $(a_0, a_1)$. Then its associated 2nd order ODE system ODE_introduction is completely integrable. In fact, denoting then are first integrals of the system. Moreover, $\xi_i,i=1\cdots n$ satisfy the following ODE system: for explicit real constants $k_1,k_2$ depending only on $a_0, a_1, c_{n-1}$ and $c_{n}$.

Theorems & Definitions (31)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3: Theorem \ref{['thm_entiren_sub']} and Theorem \ref{['thm_slag_entire']}
  • Definition 2.1
  • Proposition 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • Proposition 3.1
  • Proposition 3.2
  • ...and 21 more