Critical points of the area functional of a complex closed curve on the manifold of Kähler metrics期刊界 All Journals 搜尽天下杂志传播学术成果专业期刊搜索期刊信息化学术搜索

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

Authors:	Abel Castorena

Affiliation:	CIMAT, AP. 402, CP. 36000 Guanajuato, Gto. Mexico

Abstract:	We consider a compact complex manifold of dimension that admits Kähler metrics and we assume that is a closed complex curve. We denote by the space of classes of Kähler forms $[omega ]in H^{1,1}(M,mathbb{R} )$ that define Kähler metrics of volume 1 on and define $mathbf{A}_{C}:mathcal{KC}(1)to mathbb{R}$ by $mathbf{A}_{C}([omega ])=int_{C} omega =text{area of }Ctext{ in the induced metric by }omega$ . We show how the Riemann-Hodge bilinear relations imply that any critical point of $mathbf{A}_{C}$ is the strict global minimum and we give conditions under which there is such a critical point : A positive multiple of $[omega ]^{n-1}in H^{2n-2}(M,mathbb{R} )$ is the Poincaré dual of the homology class of . Applying this to the Abel-Jacobi map of a curve into its Jacobian, , we obtain that the Theta metric minimizes the area of within all Kähler metrics of volume 1 on .

Keywords:	K" {a}hler form, K" {a}hler manifold, Riemann-Hodge bilinear relations, Jacobian of a curve

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