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- Kähler–Einstein_metric abstract "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 Kähler metric. Therefore, the first Chern class is either negative, or zero, or positive.When the first Chern class is negative, Aubin and Yau proved that there is always a Kähler–Einstein metric.When 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.The third case, the positive or Fano case, is the hardest. In this case, there is a non-trivial obstruction to existence. In 2012, Chen, Donaldson, and Sun proved that in this case existence is equivalent to an algebro-geometric criterion called K-stability. Their proof appeared in a series of articles in the Journal of the American Mathematical Society.".
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- Kähler–Einstein_metric wikiPageWikiLink Calabi_conjecture.
- Kähler–Einstein_metric wikiPageWikiLink Calabi–Yau_manifold.
- Kähler–Einstein_metric wikiPageWikiLink Category:Differential_geometry.
- Kähler–Einstein_metric wikiPageWikiLink Chern_class.
- Kähler–Einstein_metric wikiPageWikiLink Complex_manifold.
- Kähler–Einstein_metric wikiPageWikiLink Differential_geometry.
- Kähler–Einstein_metric wikiPageWikiLink Einstein_manifold.
- Kähler–Einstein_metric wikiPageWikiLink Kähler_manifold.
- Kähler–Einstein_metric wikiPageWikiLink Kähler_metric.
- Kähler–Einstein_metric wikiPageWikiLink Manifold.
- Kähler–Einstein_metric wikiPageWikiLink Ricci-flat.
- Kähler–Einstein_metric wikiPageWikiLink Ricci-flat_manifold.
- Kähler–Einstein_metric wikiPageWikiLink Riemannian_manifold.
- Kähler–Einstein_metric wikiPageWikiLink Riemannian_metric.
- Kähler–Einstein_metric wikiPageWikiLink Shing-Tung_Yau.
- Kähler–Einstein_metric wikiPageWikiLinkText "Kähler–Einstein metric".
- Kähler–Einstein_metric wikiPageWikiLinkText "Kähler–Einstein metric".
- Kähler–Einstein_metric hasPhotoCollection Kähler–Einstein_metric.
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- Kähler–Einstein_metric subject Category:Differential_geometry.
- Kähler–Einstein_metric comment "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.".
- Kähler–Einstein_metric label "Kähler–Einstein metric".
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- Kähler–Einstein_metric sameAs Q6453393.
- Kähler–Einstein_metric sameAs Q6453393.
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