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

We show that a compact weighted extremal Kahler manifold, as defined by the third author (2019), has coercive weighted Mabuchi energy with respect to a maximal complex torus T e in the reduced group of complex automorphisms. This provides a vast extension and a unification of a number of results concerning Kahler metrics satisfying special curvature conditions, including Kahler metrics with constant scalar curvature, extremal Kahler metrics, Kahler-Ricci solitons, and their weighted ex -tensions. Our result implies the strict positivity of the weighted Donaldson-Futaki invariant of any nonproduct T e-equivariant smooth Kahler test configuration with reduced central fibre, a property known as T e-equivariant weighted K-polystability on such test configurations. It also yields the T e-uniform weighted K-stability on the class of smooth T e-equivariant polarized test configurations with reduced central fibre. For a class of fibrations constructed from principal torus bundles over a product of Hodge cscK manifolds, we use our results in conjunction with results of Chen and Cheng (2021), He (2019) and Han and Li (2022) in order to characterize the existence of extremal Kahler metrics and Calabi-Yau cones associated to the total space, in terms of the coercivity of the weighted Mabuchi energy of the fibre. This yields a new existence result for Sasaki-Einstein metrics on certain Fano toric fibrations, extending the results of Futaki, Ono and Wang (2009) in the toric Fano case, and of Mabuchi and Nakagawa (2013) in the case of Fano P 1-bundles.