Geometric Analysis Seminar —— Complex Monge-Amp\`ere equation in Orlicz space and its geometric estimates
报告人：张雷 (清华大学)
时间：2026-04-22 14:00-15:00
地点：智华楼109
【摘要】
In this talk, we establish diameter bounds and volume estimates for compact K\"ahler manifolds equipped with K\"ahler metrics or currents $\omega$, assuming the associated measure lies in a specific Orlicz space and satisfies an integrability condition. Firstly, we prove a priori estimates for solutions of the complex Monge-Amp\`ere equation in Orlicz spaces, encompassing $L^{\infty}$ and stability estimates. This is achieved by employing Ko{\l}odziej's approach and the argument of Guo-Phong-Tong-Wang, respectively. Secondly, building on the work of Guo-Phong-Song-Sturm, we derive the uniform (local/global) integral estimates of the Green's function and its gradient for the associated K\"ahler metric $\omega$. If time permitted, we will discuss the regularity of solutions to the complex Monge-Amp\`ere equation in this setting and its related geometric applications. This talk based on a joint work with my advisor Prof. Zhenlei Zhang.
