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We prove the uniqueness, up to a pull-back by an element of a suitablesubgroup of complex automorphisms, of the weighted extremal K\"ahler metrics ona compact K\"ahler manifold introduced in our previous work. This extends aresult by Berman--Berndtsson and Chen--Paun--Zeng in the extremal K\"ahlercase. Furthermore, we show that a weighted extremal K\"ahler metric is a globalminimum of a suitable weighted version of the modified Mabuchi energy. Thisimplies a suitable notion of weighted K-semistability of a K\"ahler manifoldadmitting a weighted extremal K\"ahler metric.

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