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Calabi Conjecture

The Calabi conjecture, proved by Yau (1977, 1978), states that if M is a compact Kähler manifold and rho is a real closed form of type (1,1) representing 2pic_1(M), then every cohomology class containing a Kähler form contains a unique Kähler metric omega^' whose ricci form is rho.

In particular, if the first Chern class of M vanishes, then each cohomology class containing a Kähler form contains a unique Ricci-flat Kähler metric. This special case supplies the Ricci-flat Kähler metrics used in the theory of Calabi-Yau spaces.

See also

Calabi-Yau Space, Chern Class, Kähler Manifold, Ricci Form

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References

Calabi, E. "The Space of Kähler Metrics." In Proceedings of the International Congress of Mathematicians, Amsterdam, 1954, Vol. 2. Amsterdam, Netherlands: North-Holland, pp. 206-207, 1954.Yau, S.-T. "Calabi's Conjecture and Some New Results in Algebraic Geometry." Proc. Natl. Acad. Sci. USA 74, 1798-1799, 1977. https://doi.org/10.1073/pnas.74.5.1798.Yau, S.-T. "On the Ricci Curvature of a Compact Kähler Manifold and the Complex Monge-Ampère Equation, I." Comm. Pure Appl. Math. 31, 339-411, 1978. https://doi.org/10.1002/cpa.3160310304.

Cite this as:

Weisstein, Eric W. "Calabi Conjecture." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/CalabiConjecture.html

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