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Let (X,\Omega) be a closed polarized complex manifold, g be an extremalmetric on X that represents the K\"ahler class \Omega, and G be a compactconnected subgroup of the isometry group Isom(X,g). Assume that the Futakiinvariant relative to G is nondegenerate at g. Consider a smooth family $(M \toB)$ of polarized complex deformations of (X,\Omega)\simeq (M_0,\Theta_0)provided with a holomorphic action of G with trivial action on B. Then forevery t\in B sufficiently small, there exists an h^{1,1}(X)-dimensional familyof extremal Kaehler metrics on M_t whose K\"ahler classes are arbitrarily closeto \Theta_t. We apply this deformation theory to show that certain complexdeformations of the Mukai-Umemura 3-fold admit Kaehler-Einstein metrics.

Deformation of extremal metrics, complex manifolds and the relative Futaki invariant