un 2 00 7 Remarks on the existence of bilaterally symmetric
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چکیده
The study of extremal Kähler metric is initiated by the seminal works of Calabi [4], [5]. Let (M, [ω]) be a compact Kähler manifold with fixed Kähler class [ω]. For any Kähler metrics g in the fixed Kähler class [ω], the Calabi energy C(g) is defined as C(g) = M s 2 dµ, where s is the scalar curvature of g. The extremal Kähler metric is the critical point of the Calabi energy. The Euler-Lagrange equation is ¯ ∂∇ 1,0 s = 0. In other words, Ξ = ∇ 1,0 s is a holomorphic vector field (we call it extremal vector field from now on). From PDE point of view, the existence of the extremal metric is to solve a 6th order nonlinear elliptic equation. According to Chen [6] (c.f. Donaldson [9] for algebraic case), there is a priori greatest lower bound for the Calabi energy in any fixed Kähler class. This a priori lower bound can be computed explicitly as A([ω]) = (c 1 · [ω]) 2 [ω] 2 − 1 32π 2 F (Ξ, [ω]), where F (Ξ, [ω]) is the Futaki invariant of class [ω]. Note that the extremal vector field Ξ is determined [10] up to conjugation without the assumption of the existence of an extremal metric. By E. Calabi [5], extremal Kähler metrics minimizes the Calabi energy locally. By X.X. Chen ([6]) and S.K. Donaldson ([9]), we know A([ω]) ≤ 1 32π 2 min g∈[ω] C(g), where the equality holds when there is an extremal Kähler metric in [ω].
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M ay 2 00 7 Remarks on the existence of bilaterally symmetric extremal
The study of extremal Kähler metric is initiated by the seminar work of Calabi [4], [5]. Let (M, [ω]) be a compact Kähler manifold with fixed Kähler class [ω]. The extremal Kähler metric is the critical point of the Calabi energy C(g) for any Kähler metrics g in the fixed Kähler class [ω], C(g) = M s 2 dµ, where s is the scalar curvature of g. The extremal condition asserts that ¯ ∂∇s = 0. In o...
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