Abstract. Given a (p, p) form Ω on a Hermitian manifold (M n, ω) of dimension n ≥ 2, where 1 ≤ p ＜ n is an integer, we study the problem of finding a strongly positive (p, p) form in the Apelli cohomology class of Ω with prescribed volume form. For p = 1, this is equivalent to the classical Calabi conjecture solved by S. T. Yau in the Kahler case, while for p = n - 1 it corresponds to the Gauduchon conjecture proved by Szekelyhidi-Tosatti-Weinkove more recently. From the PDE point of view, this leads to a new fully nonlinear elliptic equation which falls outside the framework developed by Cafferlli-Nirenberg-Spruck. We shall treat a general class of PDEs which also arise from other geometric problems. The talk is based on work with my student Mathew George.

嘉宾简介：关波，俄亥俄州立大学数学系教授。研究方向为非线性偏微分方程和几何分析, 主要工作包括一般区域/流形上实和复蒙日-安培方程；常高斯曲率曲面的普拉图问题；闵可夫斯基问题的推广；关于双曲空间中具有常曲率和给定渐近边界的完备曲面的研究；以及实或复流形上一般完全非线性偏微分方程。其部分学术论文发表在 Adv. Math.，Amer. J. Math.，Annals of Math.，CPAM，Duke Math. J.，JDG，J. Eur. Math. Soc.，J. Reine Angew. Math.等学术期刊上。