In differential geometry, a Kähler–Einstein metric on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric. A manifold is said to be Kähler–Einstein if it admits a Kähler–Einstein metric. The most important special case of these are the Calabi–Yau manifolds, which are Kähler and Ricci-flat.
The most important problem for this area is the existence of Kähler–Einstein metrics for compact Kähler manifolds.
In the case in which there is a Kähler metric, the Ricci curvature is proportional to the Kahler metric. Therefore, the first Chern class is either the negative, or zero, or positive.
In the first case, the hyperbolic case, in which the first Chern class is negative, Aubin and Yau proved that there is always a Kähler–Einstein metric.
In the second case, the parabolic case, in which the first Chern class is zero, Yau proved the Calabi conjecture that there is always a Kähler–Einstein metric. Shing-Tung Yau was awarded with his Fields medal because of this work. That leads to the name Calabi–Yau manifolds.
In the third case, the elliptic case, in which the first Chern class is positive, the problem is still open.
- Moroianu, Andrei (2007). Lectures on Kähler Geometry. London Mathematical Society Student Texts 69. Cambridge. ISBN 978-0-521-68897-0.
- Existence of Kähler-Einstein Metrics, a blog post.
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