Constant scalar curvature Kähler metric
In
Extremal Kähler metrics
Constant scalar curvature Kähler metrics are specific examples of a more general notion of canonical metric on Kähler manifolds, extremal Kähler metrics. Extremal metrics, as the name suggests, extremise a certain functional on the space of Kähler metrics, the Calabi functional, introduced by
Calabi functional
The Calabi functional is a functional defined on the space of
where is the
Extremal metrics
By definition, an extremal Kähler metric is a critical point of the Calabi functional.,[1] either local or global minimizers. In this sense extremal Kähler metrics can be seen as the best or canonical choice of Kähler metric on any compact Kähler manifold.
Constant scalar curvature Kähler metrics are examples of extremal Kähler metrics which are absolute minimizers of the Calabi functional. In this sense the Calabi functional is similar to the
In some circumstances constant scalar curvature Kähler metrics may not exist on a compact Kähler manifold, but extremal metrics may still exist. For example, some manifolds may admit Kähler–Ricci solitons, which are examples of extremal Kähler metrics, and explicit extremal metrics can be constructed in the case of surfaces.[3]
The absolute minimizers of the Calabi functional, the constant scalar curvature metrics, can be alternatively characterised as the critical points of another functional, the
Holomorphy potentials
There is an alternative characterization of the critical points of the Calabi functional in terms of so-called holomorphy potentials.
A holomorphy potential is a complex-valued function such that the vector field defined by is a
A Kähler metric is extremal, a minimizer of the Calabi functional, if and only if the scalar curvature is a holomorphy potential. If the scalar curvature is constant so that is cscK, then the associated holomorphy potential is a constant function, and the induced holomorphic vector field is the zero vector field. In particular on a Kähler manifold which admits no non-zero holomorphic vector fields, the only holomorphy potentials are constant functions and every extremal metric is a constant scalar curvature Kähler metric.
The existence of constant curvature metrics are intimately linked to obstructions arising from holomorphic vector fields, which leads to the
See also
- Mabuchi functional
- Kähler-Einstein metric
References
- Biquard, Olivier (2006), "Métriques kählériennes à courbure scalaire constante: unicité, stabilité", Astérisque, Séminaire Bourbaki. Vol. 2004/2005 Exp. No. 938 (307): 1–31, MR 2296414
- Donaldson, S. K. (2001), "Scalar curvature and projective embeddings. I", Journal of Differential Geometry, 59 (3): 479–522, MR 1916953
- Donaldson, S. K. (2002), "Scalar curvature and stability of toric varieties", Journal of Differential Geometry, 62 (2): 289–349, MR 1988506
- ^ a b Calabi, E., 1982. EXTREMAL KAHLER METRICS. In SEMINAR ON DIFFERENTIAL GEOMETRY (p. 259).
- ^ a b Székelyhidi, G., 2014. An Introduction to Extremal Kahler Metrics (Vol. 152). American Mathematical Soc.
- MR 1647571
- ^ Gauduchon, Paul. 2014. Calabi's extremal Kähler metrics: An elementary introduction