In mathematics, the Calabi conjecture was a conjecture about the existence of certain "nice" Riemannian metrics on certain complex manifolds, made by Eugenio Calabi (1954, 1957) and proved by Shing-Tung Yau (1977, 1978). Yau received the Fields Medal in 1982 in part for this proof.
The Calabi conjecture states that a compact Kähler manifold has a unique Kähler 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 Kähler metric in the same class with vanishing Ricci curvature; these are called Calabi–Yau manifolds.
More formally, the Calabi conjecture states:
- If M is a compact Kähler manifold with Kähler metric and Kähler form, and R is any (1,1)-form representing the manifold's first Chern class, then there exists a unique Kähler metric on M with Kähler form such that and represent the same class in cohomology H2(M,R) and the Ricci form of is R.
The Calabi conjecture is closely related to the question of which Kähler manifolds have Kähler–Einstein metrics.
Read more about Calabi Conjecture: Kähler–Einstein Metrics, Outline of The Proof of The Calabi Conjecture
Famous quotes containing the word conjecture:
“There is something fascinating about science. One gets such wholesale returns of conjecture out of such a trifling investment of fact.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)