Abstract
We show in this paper by applying the Seiberg–Witten theory developed by Taubes and LeBrun that a compact almost Kähler–Einstein 4-manifold of negative scalar curvature s is Kähler–Einstein if and only if the L2-norm satisfies ∫Ms2dv=32π2(2χ+3τ)(M). The Einstein condition can be weakened by the topological condition (2χ+3τ)(M)>0.
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