Uniqueness of Einstein Kähler Metrics Modulo Connected Group Actions

Abstract

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".