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 equivale...

We show that some embedded standard $13$-spheres in Shimada's exotic $15$-spheres have $\mathbb{Z}_2$ quotient spaces, $P^{13}$s, are fake real $13$-dimensional projective i.e., they homotopy equivalent, but not diffeomorphic to the $\mathbb{R}\mathrm{P}^{13}$. As observed by F. Wilhelm and second named author [RW], Davis $\mathsf{SO}(2)\times \mathsf{G}_2$ actions on descend cohomogeneity one ...