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...

on Kähler manifolds, such as the existence of special connections like Hermitian Yang–Mills connections, or special metrics such as Kähler–Einstein metrics...

such metrics on K3, 57 parameters of which give rise to Einstein metrics which are not related by isometries or rescalings. Kähler–Einstein metrics exist...

exist a stability condition which characterises the existence of a Kähler–Einstein metric on a Fano manifold. It was defined in reference to the K-energy...

Donaldson-Uhlenbeck-Yau theorem, that existence of a Kähler-Einstein metric should correspond to stability of the underlying Kähler manifold in a certain sense of geometric...

curvature Kähler metrics (cscK metrics). In 1954, Eugenio Calabi formulated a conjecture about the existence of Kähler metrics on compact Kähler manifolds...

Kähler metric (cscK metric) is a Kähler metric on a complex manifold whose scalar curvature is constant. A special case is a Kähler–Einstein metric,...

functional on the space of Kähler potentials of a compact Kähler manifold whose critical points are constant scalar curvature Kähler metrics. The Mabuchi functional...

Hyper-Kähler manifold Kähler quotient Hyperkähler quotient Kähler–Einstein metric Nearly Kähler manifold Quaternion-Kähler manifold Special Kähler geometry...

The Levi-Civita connection of a Kähler–Einstein metric is Hermite–Einstein with respect to the Kähler–Einstein metric. (These examples are however dangerously...

the entire manifold, and many special metrics such as constant scalar curvature metrics and Kähler–Einstein metrics are constructed intrinsically using...

Gromov–Hausdorff limits. The summary of the existence proof for Kähler–Einstein metrics appears in Chen, Donaldson & Sun (2014). Full details of the proofs...

there is exactly one Kähler metric in each Kähler class whose Ricci form is R. (Some compact complex manifolds admit no Kähler classes, in which case...

Weil–Petersson metric is Kähler but it is not complete. Cheng and Yau showed that there is a unique complete Kähler–Einstein metric on Teichmüller space....

In differential geometry, a quaternion-Kähler manifold (or quaternionic Kähler manifold) is a Riemannian 4n-manifold whose Riemannian holonomy group is...

well as the Yau–Tian–Donaldson conjecture about the existence of Kähler–Einstein metrics on Fano varieties, and the Thomas–Yau conjecture about existence...

Kähler geometry, the situation is not as well understood. A four-dimensional closed and oriented manifold supporting any Einstein Riemannian metric must...

question of how Kähler metrics on compact complex manifolds of nonpositive first Chern class can be deformed into Kähler–Einstein metrics.[Y78a] Akito Futaki...

Hamilton–Jacobi–Einstein equation Higher-dimensional Einstein gravity Kähler–Einstein metric Wiener–Khinchin–Einstein theorem Einstein pseudotensor Stark–Einstein law...

series, "Kähler–Einstein metrics on Fano manifolds, I, II and III". Donaldson, Simon; Sun, Song (2014). "Gromov-Hausdorff limits of Kähler manifolds...

class. M {\displaystyle M} has a Kähler metric with vanishing Ricci curvature. M {\displaystyle M} has a Kähler metric with local holonomy contained in...

}}\right|_{\tau =0}.} Jordan curve Kähler-Einstein metric Kähler metric Killing vector field Length metric the same as intrinsic metric. Levi-Civita connection is...

correspondence, and existence results for Kähler–Einstein metrics and constant scalar curvature Kähler metrics. These results often feed back into complex...

proposal for finding Kähler metrics of constant scalar curvature.[C82a] More broadly, Calabi introduced the notion of an extremal Kähler metric, and established...

with Yau, he also showed that Kähler manifolds with negative first Chern classes always admit Kähler–Einstein metrics, a result closely related to the...

Partly for this reason, the Einstein field equations propose that spacetime can be described by a pseudo-Riemannian metric, with a strikingly simple relationship...

Fubini–Study metric (IPA: /fubini-ʃtuːdi/) is a Kähler metric on a complex projective space CPn endowed with a Hermitian form. This metric was originally...

function. Cao, Huai Dong (1985). "Deformation of Kähler metrics to Kähler–Einstein metrics on compact Kähler manifolds". Inventiones Mathematicae. 81 (2):...

three-article series, "Kähler–Einstein metrics on Fano manifolds, I, II and III". Chen, Xiuxiong. The space of Kähler metrics. J. Differential Geom. 56...

differential geometry (Kähler manifolds), including the existence of canonical metrics (such as extremal Kähler and Kähler-Einstein metrics) on projective manifolds...