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Yau's conjecture, proposed by mathematician Shing-Tung Yau, relates to the study of Kähler manifolds, particularly in the context of complex differential geometry and algebraic geometry. Specifically, it addresses the existence of Kähler metrics with specific curvature properties on complex manifolds. One of the notable forms of Yau's conjecture is concerned with the existence of Kähler-Einstein metrics on Fano manifolds.

Yau's conjecture refers to a prediction made by the mathematician Shing-Tung Yau regarding the first eigenvalue of the Laplace operator on compact Riemannian manifolds. Specifically, the conjecture addresses the relationship between the geometry of a manifold and the spectrum of the Laplace operator defined on it.