2025 Theses Doctoral

On the method of auxiliary Monge-Ampère equations in complex geometry

Wang, Chuwen

A priori estimates, especially the 𝐿^∞ estimates, are the core to solving partial differential equations (PDE) arising from complex geometry. It has been a longstanding question whether one can find a PDE method to prove a 𝐿^∞ estimate of the complex Monge-Ampère equation with a singular right-hand side. Recently, the method of auxiliary Monge-Ampère equations developed by B. Guo, D. H. Phong, and F. Tong has emerged to be a major advance regarding this question, with immense generality.

We use this method to prove the 𝐿^∞ estimate for complex Monge-Ampère and Hessian questions on a nef cohomology class, and for the (𝑛 - 1)-form equations on Hermitian manifolds. We also prove a sharp uniform modulus of continuity estimate for complex Monge-Ampère equations, together with its geometric application on diameter bounds for Kähler metrics.