In this paper, we prove that on a smooth K\"ahler manifold, the G-coercivity of the weighted Mabuchi functional implies the existence of weighted-cscK (extremal) metric (firstly studied in {Lah19}), e.g, cscK metric, K\"ahler--Ricci soliton. Furthermore, we generalize this result to the case of G-equivariant Q-Gorenstein smoothable projective klt varieties.
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Han et al. (2024) studied this question.
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