The Critical LYZ Equation in K\"ahler Geometry

TL;DR

This paper proves the existence of smooth solutions to the LYZ equation at the critical phase in Kähler geometry, resolving a longstanding problem and applying results to specific Hessian equations under weaker conditions.

Contribution

It establishes the critical case solvability of the LYZ equation, advancing understanding in Kähler geometry and Hessian equations.

Findings

Existence of smooth solutions at the critical phase for the LYZ equation.

Solution of the 3D Hessian equation σ₂=1 under weaker assumptions.

Solution of the 4D Hessian quotient equation σ₃=σ₁ with relaxed conditions.

Abstract

We establish the existence of smooth solutions for the LYZ equation at the critical phase $\theta =(n-2)\frac{\pi}{2}$, thereby solving the critical case of a problem posed by Collins-Jacob-Yau and Li concerning the solvability for phase $\theta \leq (n-2)\frac{\pi}{2}$. As applications, we solve the 3D Hessian equation $\sigma_2 = 1$ and the 4D Hessian quotient equation $\sigma_3 = \sigma_1$ under weaker assumptions than previously required.