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Lemma 6
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Let ( X , ω ) (X,\omega) be a compact Kähler manifold of complex dimension n n , let h h be a smooth function such that ∫ X ω n = ∫ X e h ω n \int_{X}\omega^{n}=\int_{X}e^{h}\omega^{n} and φ ∈ 𝒫 ω \varphi\in{\cal P}_{\omega} a solution of the complex Monge-Ampère equation
( ω + i ∂ ∂ ¯ φ ) n = e h + λ φ ω n , \displaystyle(\omega+i\partial\bar{\partial}\varphi)^{n}=e^{h+\lambda\varphi}\omega^{n}\,,
(4.5)
λ > 0 \lambda>0 . Consider also two solutions φ ′ φ ′′ ∈ 𝒫 ω \varphi^{\prime}\,\varphi^{\prime\prime}\in{\cal P}_{\omega} of the complex Monge-Ampère equation ( ω + i ∂ ∂ ¯ φ ) n = e h ω n (\omega+i\partial\bar{\partial}\varphi)^{n}=e^{h}\omega^{n} such that min X φ ′ = 0 = max X φ ′′ \min_{X}\varphi^{\prime}=0=\max_{X}\varphi^{\prime\prime} . Then φ ′′ ≤ φ ≤ φ ′ \varphi^{\prime\prime}\leq\varphi\leq\varphi^{\prime} .