Convexity of the weighted Mabuchi functional and the uniqueness of weighted extremal metrics

We prove the uniqueness, up to a pull-back by an element of a suitable subgroup of complex automorphisms, of the weighted extremal Kahler metrics on a compact Kahler manifold introduced in our previous work. This extends a result by Berman--Berndtsson and Chen--Paun--Zeng in the extremal Kahler case. Furthermore, we show that a weighted extremal Kahler metric is a global minimum of a suitable weighted version of the modified Mabuchi energy. This implies a suitable notion of weighted K-semistability of a Kahler manifold admitting a weighted extremal Kahler metric.