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The Kahler-Einstein metric is an example of a canonical metric on a Kahler manifold. We say that a metric $\omega$ is Kahler-Einstein if $Ric(\omega)=\lambda\omega$, where $\lambda\in\{-1,0,+1\}$.

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Does there exists a complete Kahler metric with positive scalar curvature on Poincare disk?

Let (M,g) be a Poincare disk. Does there exists a complete Kahler metric h, such that the scalar curvature of h is positive?
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