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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 Kähler metrics on a compact Kähler manifold introduced in our previous work. This extends a result by Berman--Berndtsson and Chen--Paun--Zeng in the extremal Kähler case. Furthermore, we show that a weighted extremal Kähler 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 Kähler manifold admitting a weighted extremal Kähler metric.