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We show that a compact weighted extremal Kahler manifold (as defined by thethird named author) has coercive weighted Mabuchi energy with respect to amaximal complex torus in the reduced group of complex automorphisms. Thisprovides a vast extension and a unification of a number of results concerningKahler metrics satisfying special curvature conditions, including constantscalar curvature Kahler metrics, extremal Kahler metrics, Kahler-Ricci solitonsand their weighted extensions.  Our result implies the strict positivity of the weighted Donaldson-Futakiinvariant of any non-product equivariant smooth K\"ahler test configurationwith reduced central fibre, a property also known as weighted K-polystabilityon such test configurations. For a class of fibre-bundles, we use our result inconjunction with the recent results of Chen-Cheng, He, and Han-Li in order tocharacterize the existence of extremal Kahler metrics and Calabi-Yau conesassociated to the total space, in terms of the coercivity of the weightedMabuchi energy of the fibre. In particular, this yields an existence result forSasaki-Einstein metrics on Fano toric fibrations, extending the results ofFutaki-Ono-Wang in the toric Fano case, and of Mabuchi-Nakagawa in the case ofFano projective line bundles.

Weighted K-stability and coercivity with applications to extremal Kahler and Sasaki metrics