UBC DG-MP-PDE Seminar: Huang Liding

The Calabi-Yau theorem says that given any smooth representative Φ of the first Chern class, there exists a unique K\”ahler metric ω cohomologous to α such that Ricci(ω)=Φ. It is natural to investigate whether similar results hold when the manifolds is non-K\”ahler. In this talk, we will introduce the form type Calabi-Yau equation which can used to prove a version of the Calabi conjecture for balanced metrics. We define the astheno-Ricci curvature and prove that there exists a solution for the form type Calabi-Yau equation if the astheno-Ricci curvature is non-positive.