A remark on extremal Kähler metrics (Q1081136)

It is known that Kähler-Einstein metrics on Kähler manifolds are solutions of a certain variational problem, introduced by Calabi. Calabi also proved that for such manifolds with extremal metrics the group of holomorphic automorphisms must contain a nontrivial compact real Lie subgroup, if it has positive dimension. The author constructs two manifolds which fail to have such a compact subgroup of their automorphism group; so these are examples of manifolds which do not have extremal metrics.