The Yau–Tian–Donaldson Conjecture for Cohomogeneity One Manifolds

We prove the Yau–Tian–Donaldson conjecture for cohomogeneity one manifolds, that is, projective manifolds equipped with a holomorphic action of compact Lie group at least real hypersurface orbit. Contrary to what seems be popular belief, such do not admit extremal Kähler metrics in all classes general. More generally, we rank polarized spherical varieties, $$G$$ -uniform K-stability is equivalent respect special -equivariant test configurations. This furthermore encoded by single combinatorial condition, checkable practice. illustrate on examples and answer along way question Kanemitsu.