On the existence of weighted-cscK metrics

In this paper, we prove that on a smooth K\"ahler manifold, the $\mathbb{G}$-coercivity of the weighted Mabuchi functional implies the existence of the (v, w)-weighted-cscK (extremal) metric with v log-concave (firstly studied in \cite{Lah19}), e.g, cscK metrics, K\"ahler-Ricci solitons, $\mu$-cscK metrics.