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We consider a compact complex manifold M of dimension n that admits Kahler metrics and we assume that C → M is a closed complex curve. We denote by KC(1) the space of classes of Kahler forms [ω] E H 1,1 (M,R) that define Kahler metrics of volume 1 on M and define A C : KC(1) → R by A C ([ω]) = ∫ C w = area of C in the induced metric by ω. We show how the Riemann-Hodge bilinear relations imply that any critical point of A C is the strict global minimum and we give conditions under which there is such a critical point [ω]: A positive multiple of [ω] n-1 ∈ H 2n-2 (M,R) is the Poincare dual of the homology class of C. Applying this to the Abel-Jacobi map of a curve into its Jacobian, C → J(C), we obtain that the Theta metric minimizes the area of C within all Kahler metrics of volume 1 on J(C).

Critical points of the area functional of a complex closed curve on the manifold of Kähler metrics