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...

This paper is the first of a series of papers in which we generalize our results in (Asian J. of Math. 4, 817–830 (2000); J. Geom. Anal. 12, 63–79 (2002); Intern. J. Math. 14, 259–287 (2003)) to the general complex compact almost homogeneous manifolds of real cohomogeneity one. In this paper we deal with the exceptional case of the G2 action (Cf. Intern. J. Math. 14, 259–287 (2003), p. 285). In...

We prove an existence result for the prescribed Ricci curvature equation certain doubly warped product metrics on $$\mathbb {S}^{d_1+1}\times \mathbb {S}^{d_2}$$ , where $$d_i \ge 2$$ . If T is a metric satisfying assumptions, we show that can be scaled independently two factors so as to itself tensor of some metric. This first global problem closed cohomogeneity one manifolds since work Hami...

We construct all possible complete metrics of cohomogeneity one G2 holonomy with S × S principal orbits from gauged supergravity. Our approach rests on a generalization of the twisting procedure used in this framework. It corresponds to a non-trivial embedding of the special Lagrangian three–cycle wrapped by the D6–branes in the lower dimensional supergravity. There are constraints that neatly ...