2026 Theses Doctoral

Fully Nonlinear Partial Differential Equations on Hermitian Manifolds

Liang, Shuang

This thesis studies fully nonlinear partial differential equations on Hermitian manifolds, with emphasis on a priori estimates and geometric applications. It consists of two main parts.

In the first part, we obtain a priori 𝐿^∞ estimates for a general class of (𝑛 − 1)-form fully nonlinear partial differential equations on compact Hermitian manifolds. Our method relies on the local version of the comparison with auxiliary Monge-Ampère equations, developed earlier by B. Guo and D. H. Phong. The key is to find the appropriate elliptic operator such that the maximum principle applies. This is joint work with N. Klemyatin and C. Wang.

In the second part, we study the continuity equation for Hermitian metrics, introduced by La Nave--Tian and extended to the Hermitian setting by Sherman--Weinkove. We prove local Calabi and higher order estimates for solutions to the continuity equation and apply them to show that on a compact complex manifold, the Chern scalar curvature of a solution must blow up at a finite-time singularity. Additionally, starting from certain classes of initial data on Oeljeklaus--Toma manifolds, we prove Gromov--Hausdorff and smooth convergence of the metric to a particular non-negative (1,1)-form as 𝑡 → ∞. This is joint work with X. S. Shen and K. Smith.