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Kähler-Einstein Metrics With Non-positive Scalar Curvature

G. Tian
Published 2000 · Mathematics

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We have seen that the Ricci curvature represents the first Chern class. In this section, we will consider the converse problem, namely, given a Kahler class [ω] ∈ H2 (M, ℝ) ∩ H1,2 (M, ℂ) on a compact Kahler manifold M and any form Ω representing the first Chern class, can we find a metric ω ∈ [ω] such that Ric(ω) = Ω? This is known as the Calabi conjecture and it was solved by Yau in 1976. We will state it here as a theorem and refer to it as the Calabi-Yau Theorem.

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