Xiao Xia (UQAM)

Title: On Weighted Twisted K-Energy and Its Applications.

Abstract: I will discuss the convexity of the twisted Mabuchi K-energy along weak geodesics in the finite-energy space E^1. This functional is a generalization of the classical Mabuchi K-energy (associated to the constant scalar curvature Kähler problem, cscK in short) and allows us to deal with mixed cusp and conic singularities. I will then explain why coercivity of the twisted Mabuchi K-energy is an open condition with respect to perturbations of the cone angle. As applications, this gives openness for the existence of cscK cone metrics and shows that coercivity at the cusp limit implies the existence of cscK cone metrics for small cone angles. If time permits, I will also briefly indicate how these results extend to more general geometric PDEs.