M ay 1 99 9 EXTREMAL KÄHLER METRICS AND RAY - SINGER ANALYTIC TORSION

Let (X, [ω]) be a compact Kähler manifold with a fixed Kähler class [ω]. Let K ω be the set of all Kähler metrics on X whose Kähler class equals [ω]. In this paper we investigate the critical points of the functional g ∈ K ω → Q(g) = v g T 0 (X, g) 1/2 , where v is a fixed nonzero vector of the determinant line λ(X) associated to H * (X) and T 0 (X, g) is the Ray-Singer analytic torsion. For a polarized algebraic manifold (X, L) we consider a twisted version Q L (g) of this functional and assume that c 1 (L) = [ω]. Then the critical points of Q L are exactly the metrics g ∈ K ω of constant scalar curvature. In particular, if c 1 (X) = 0 or if c 1 (X) < 0 and 1 2π [ω] = −c 1 (X), then K ω contains a unique Kähler-Einstein metric g KE and Q L attains its absolut maximum at g KE .