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

curvature is K = −1/r2. The scalar curvature is thus S = −n(n − 1)/r2. The scalar curvature is also constant when given a Kähler metric of constant holomorphic...

Kähler–Einstein metric, it is possible to study mild generalizations including constant scalar curvature Kähler metrics and extremal Kähler metrics. When a Kähler–Einstein...

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...

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

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

Fubini–Study metric, have k = 2 n + 2. {\displaystyle k=2n+2.} Calabi–Yau manifolds admit an Einstein metric that is also Kähler, with Einstein constant k = 0...

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

equations with cosmological constant. On a Kähler manifold X {\displaystyle X} , the Ricci curvature determines the curvature form of the canonical line...

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

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...

scalar curvature equaling zero. From the definition of the Weyl curvature tensor, it is direct to see that any Ricci-flat metric has Weyl curvature equal...

given by constant scalar curvature Kähler metrics, can be interpreted as the log-norm functional for a Quillen metric on the space of Kähler metrics. Quillen...

} Jordan curve Kähler-Einstein metric Kähler metric Killing vector field Koszul Connection Length metric the same as intrinsic metric. Length space Levi-Civita...

to the fact that Cheng and Yau only considered Kähler−Einstein metrics with negative scalar curvature. The more subtle question, where Fefferman's earlier...

Kähler–Einstein metric and a holomorphic contact structure is actually the twistor space of a quaternion-Kähler manifold of positive scalar curvature...

scalar curvature. Teichmüller space also carries a complete Kähler metric of bounded sectional curvature introduced by McMullen (2000) that is Kähler-hyperbolic...

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

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...

the existence of "extremal" Kähler metrics, typically those with constant scalar curvature (see for example cscK metric). Donaldson obtained results...

with the Riemannian metric, is called quaternion-Kähler symmetric space. An irreducible symmetric space G / K is quaternion-Kähler if and only if isotropy...

defined on the entire bundle. Computing the scalar curvature of this bundle metric, one finds that it is constant on each fiber: this is the "Kaluza miracle"...

Dervan, Ruadhaí (2016). "Uniform Stability of Twisted Constant Scalar Curvature Kähler Metrics". International Mathematics Research Notices. 2016 (15):...

manifold. Kähler metrics remain Kähler under Ricci flow, and so Ricci flow has also been studied in this setting, where it is called Kähler–Ricci flow...

is a Kähler manifold, and it is characterised by being the only simply connected Kähler manifold whose holomorphic sectional curvature is constant equal...

smooth K-polystable projective variety should admit a constant scalar curvature Kähler metric. This generalisation of the conjecture for Fano manifolds...

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...

metrics of positive scalar curvature cannot exist on such manifolds. A particular consequence is that the torus cannot support any Riemannian metric of...

no. 1, 1 - 123. Chen, Xiuxiong; Cheng, Jingrui. On the constant scalar curvature Kähler metrics (I)—A priori estimates. J. Amer. Math. Soc. 34 (2021),...

and metric geometry and list of Lie group topics. List of curves topics Frenet–Serret formulas Curves in differential geometry Line element Curvature Radius...