Invariant hyperkähler metrics with a homogeneous complex structure

Abstract

We give a hyperkähler analogue of the classification of compact homogeneous Kähler manifolds. Namely we show that, for a compact semisimple Lie group G, the complete G-invariant hyperkähler manifolds which are also locally G^c-homogeneous with respect to one of the complex structures are precisely the hyperkähler metrics of Kronheimer, Biquard and Kovalev on coadjoint semisimple orbits of G^c. As a consequence the only complete hyperkähler metrics which are of cohomogeneity one with respect to a triholomorphic action of compact semisimple Lie group are the flat metric on ℍⁿ and the Calabi metric on T*[Copf]Pⁿ.