Meaning of **Canonical metric** on complex manifolds

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What is the meaning of Canonical metric on complex manifolds ?

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For a compact Kahler manifold $M$, canonical metric can mean: (i) Kahler-Enstein metric or (ii) metric with constant scalar curvature which is the extremal metric of the Calabi functional $\displaystyle\int_M R^2$.

You can find more details from the book Canonical Metrics in Kahler Geometry by by Gang Tian.