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The Critical LYZ equation in Kähler Geometry

Jixiang Fu, Shing-Tung Yau, Dekai Zhang

TL;DR

The article resolves the critical LYZ equation θω(χ_u)=(n-2)π/2 on compact Kähler manifolds by perturbing to a family of supercritical equations, preserving a subsolution, and obtaining uniform a priori estimates. It establishes a uniform Hou–Ma–Wu–type complex Hessian bound and a gradient bound via a Liouville-type argument, enabling passage to the critical limit and yielding a unique smooth solution. The work also extends to concrete low-dimensional cases, solving the 3D 2-Hessian equation and the 4D Hessian quotient under weaker hypotheses than prior results. Overall, it advances existence theory for fully nonlinear complex Hessian equations in the critical regime and broadens applicability to curvature-type equations in Kähler geometry.

Abstract

We establish the existence of smooth solutions for the LYZ equation at the critical phase $θ=(n-2)\fracπ{2}$, thereby solving the critical case of a problem posed by Collins-Jacob-Yau and Li concerning the solvability for phase $θ\leq (n-2)\fracπ{2}$. As applications, we solve the 3D Hessian equation $σ_2 = 1$ and the 4D Hessian quotient equation $σ_3 = σ_1$ under weaker assumptions than previously required.

The Critical LYZ equation in Kähler Geometry

TL;DR

The article resolves the critical LYZ equation θω(χ_u)=(n-2)π/2 on compact Kähler manifolds by perturbing to a family of supercritical equations, preserving a subsolution, and obtaining uniform a priori estimates. It establishes a uniform Hou–Ma–Wu–type complex Hessian bound and a gradient bound via a Liouville-type argument, enabling passage to the critical limit and yielding a unique smooth solution. The work also extends to concrete low-dimensional cases, solving the 3D 2-Hessian equation and the 4D Hessian quotient under weaker hypotheses than prior results. Overall, it advances existence theory for fully nonlinear complex Hessian equations in the critical regime and broadens applicability to curvature-type equations in Kähler geometry.

Abstract

We establish the existence of smooth solutions for the LYZ equation at the critical phase , thereby solving the critical case of a problem posed by Collins-Jacob-Yau and Li concerning the solvability for phase . As applications, we solve the 3D Hessian equation and the 4D Hessian quotient equation under weaker assumptions than previously required.

Paper Structure

This paper contains 19 sections, 21 theorems, 154 equations.

Table of Contents

  1. Introduction
  2. Previous Results
  3. Our main results
  4. preliminaries
  5. The $k$-Hessian operators and the special Lagrangian operator
  6. $C^0$-estimates
  7. Complex Hessian estimates
  8. The linearized operator
  9. Key inequalities from the subsolution condition
  10. The complex Hessian estimate of Hou-Ma-Wu type
  11. Gradient estimates and the proof of the main theorems
  12. Viscosity solutions
  13. Weak solutions in the sense of currents
  14. A Liouville theorem to the equation $\sigma_{n}+\sigma_{n-1}=0$ in $\mathbb{C}^n$
  15. Gradient estimates
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $(M,\omega)$ be a compact Kähler manifold and $\chi$ a closed real (1,1)-form such that the principal argument of $\int_{M}(\chi+\sqrt{-1} \omega)^n$ is $\pi$. If there exists a smooth function $\underline{u}$ satisfying subsolution, then the critical LYZ equation LYZ1 admits a unique smooth sol

Theorems & Definitions (40)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Corollary 1.1
  • Corollary 1.2
  • Lemma 2.1
  • Lemma 2.2: Yuan yuan2006pams, Wang--Yuan wangyuan2014ajm
  • Lemma 2.3
  • proof
  • Lemma 2.4: Collins--Jacob--Yau cjy2020
  • ...and 30 more