M ay 2 00 7 Remarks on the existence of bilaterally symmetric extremal
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چکیده
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 other words, ∇s is a holomorphic vector field. From PDE point of view, the existence of the extremal metric is to solve a 6th order nonlinear elliptic equation. In this short note we show that the existence of bilaterally symmetric ex-tremal Kähler metrics on CP 2 ♯2CP 2 by following the same scheme and the same idea of [8]. The observation is that ([8] Proposition 26) still holds for A([ω]) < 9. By [6], A([ω]) gives an a priori lower bound for the Calabi energy (c.f. [9] for algebraic case). Follow the scheme in ([8]), actually we show that Theorem 0.1. For any x ∈ (0, ∞), let [ω] x = (1 + x)(F 1 + F 2) − xE denote the Kähler class of on M = CP 2 ♯2CP 2 , then there is an extremal metric in [ω] x for any x ∈ (0, ∞).
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un 2 00 7 Remarks on the existence of bilaterally symmetric
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-Lagrang...
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