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arXiv · 2511.21492

The Critical LYZ Equation in K\"ahler Geometry

Jixiang Fu·Shing-Tung Yau·Dekai Zhang
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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.

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Jixiang Fu, Shing-Tung Yau, Dekai Zhang. 2025-11-26. The Critical LYZ Equation in K\"ahler Geometry. https://arxiv.org/abs/2511.21492

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