1999/04/30 by Werner Mueller, Katrin Wendland
Mathematics · #math.DG #math.FA #msc:58G26 #msc:58E11 #msc:53C55
published as Geometric aspects of partial differential equations (Roskilde, 1998), 135--160, Contemp. Math., 242, Amer. Math. Soc., Providence, RI, 1999 · 33 pages, LaTeX2e, to be published in Geometric Aspects of partial differential equations, Proceedings of the Roskilde Conference, Sept. 1998, AMS series Contemporary Mathematics
arxiv created 1999/05/06 · arxiv updated 2009/11/30
Let (X,[ω]) be a compact Kaehler manifold with a fixed Kaehler class [ω]. Let Kωbe the set of all Kaehler metrics on X whose Kaehler class equals [ω]. In this paper we investigate the critical points of the functional Q(g)= |v|g T0(X,g)1/2 for g ∈ Kω, where v is a fixed nonzero vector of the determinant line λ(X) associated to H^*(X) and T0(X,g) is the Ray-Singer analytic torsion. For a polarized algebraic manifold (X,L) we consider a twisted version QL(g) of this functional and assume that c1(L)=[ω]. Then the critical points of QL are exactly the metrics g∈ Kωof constant scalar curvature. In particular, if c1(X)=0 or if c1(X)<0 and 1/(2π)[ω] = -c1(X), then Kωcontains a unique Kaehler-Einstein metric gKE and QL attains its absolut maximum at gKE.