On Weighted Twisted K-Energy and Its Applications

We establish the convexity of the weighted twisted Mabuchi K-energy functional along geodesics in the finite energy space $\mathcal{E}^{1,T}(X,\omega)$, covering the case of divisors with mixed cusp and conic singularities. We then prove that coercivity (relative to the complex torus) of this functional is an open condition under cone angle perturbations. This is obtained from a general result of independent interest, which shows the stability of the coercivity under perturbations by certain twist currents. In particular, this yields the openness for the existence for cscK cone metrics and proves that coercivity at the cusp limit implies existence of cscK cone metrics for small cone angles.