B

Abstract

Let M be a compact Kählerm-manifold that has a Kähler metric ds2 = gαβ̄ dz α ⊗ dz̄ . Then it is known that, for the Ricci curvature tensor Rαβ̄ = −(∂2/∂zα∂z̄β) log det(gλμ̄), √−1 2π Rαβ̄ dz α ∧ dz̄ is a closed(1,1)-form and its cohomology class is equal to the first Chern class C1(M). Conversely, it was Calabi who asked if, for any closed (1,1)-form √−1 2π R̃αβ̄ dz α ∧ dz̄ that is cohomologous toC1(M), can one find a Kähler metric̃ gαβ̄ onM such thatR̃αβ̄ is the Ricci curvature tensor of̃ gαβ̄? As a consequence of Aubin and Yau’s results, one can find a Kähler–Einstein metric on M with C1(M) = 0 orC1(M) < 0. WhenC1(M) > 0, the space of Kähler–Einstein metrics are invariant under automorphism group. However, the existence does not always hold in general [F; M; T; TY]. Instead of the Kähler–Einstein metric, we consider the notion of extremal metrics due to Calabi [C1]. Namely, fix a Kähler class 0 = [ω0] on a compact Kähler manifoldM and denote byH 0 the space of all Kähler metrics with the same fixed Kähler class 0. Now consider the functional 8 : H 0 → R, 8(g) = ∫