Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, the Calabi conjecture was a conjecture about the existence of good metrics on complex manifolds, made by Calabi in about 1954. The conjecture was proven by Shing-Tung Yau in 1976. The Calabi conjecture states that a compact Kahler manifold has a unique Kahler metric in the same class whose Ricci form is any given 2-form representing the first Chern class. In particular if the first Chern class vanishes there is a unique Kahler metric in the same class with vanishing Ricci curvature; these are called Calabi–Yau manifolds. The Calabi conjecture is closely related to the question of which Kahler manifolds have Kahler–Einstein metrics.

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