Relative uniform K-stability over models implies existence of extremal metrics on polarised smooth complex projective varieties.
Hisamoto: Stability and coercivity for toric polarizations
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A survey summarizing that existence of coupled Kähler-Einstein metrics and weighted solitons on Fano manifolds equates to generalized K-polystability conditions.
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Relative uniform $K$-stability over models implies existence of extremal metrics
Relative uniform K-stability over models implies existence of extremal metrics on polarised smooth complex projective varieties.
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On coupled K\"ahler-Einstein metrics and weighted solitons on Fano manifolds
A survey summarizing that existence of coupled Kähler-Einstein metrics and weighted solitons on Fano manifolds equates to generalized K-polystability conditions.