Throughout this paper, we fix an arbitrary n-dimensional compact complex manifold X with positive first Chern class cl(Xh>O. We then put f: the set of all Kahler forms on X representing 27!'c1(X)R' f+: ={w E f I w has positive definite Ricci tensor}, 0":={w E f Iw is an Einstein form}, C"'(Xh: the space of real-valued COO-functions on X, Aut (X): the group of holomorphic automorphisms of X, G:=AutO(X): the identity component of Aut (X).Furthermore, Aut (X) is always assumed to act from the right on f by (w, g) E fxAut(X)...-+g*w E f.The main purpose of this paper is to prove the uniqueness of Einstein Kahler metrics, if any, on X up to G-action. Such uniqueness was known only for i) Kahler C-spaces (cf. Matsushima [12]) and ii) some nonhomogeneous Einstein manifolds recently discovered by Sakane [13]. Now, the correct statement we obtain has the following stronger form as announced earlier in [9]: Theorem A. Fix an element WI of f. Let fl + : f + -+ R be the restriction to f+ of the f-energy map wE f...-+M(w" w) E R of the Kahler manifold (X, WI) (see Section 1, also [9]). Assume that 0" *9. Then (i) fl+ is boundedfrom below and takes its absolute minimum exactly on 0".

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Recently, Donaldson proved asymptotic stability for a polarized algebraic manifold M with polarization class admitting a Kähler metric of constant scalar curvature, essentially when the linear algebraic part H of Aut 0 (M ) is semisimple. The purpose of this paper is to give a generalization of Donaldson's result to the case where the polarization class admits an extremal Kähler metric, even when H is not semisimple.

For an integral Kähler class on a compact connected complex manifold, an extremal Kähler metric, if any, in the class is unique up to the action of Aut 0(M). This generalizes a recent result of Donaldson (see [4] for cases of metrics of constant scalar curvature) and that of Chen [3] for c1(M)≤0.