The Critical LYZ Equation in Kähler Geometry

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 $\theta \le (n-2)\frac{\pi }{2}$. 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.