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Subject Item
dbr:Mabuchi_functional
rdfs:label
Mabuchi functional
rdfs:comment
In mathematics, and especially complex geometry, the Mabuchi functional or K-energy functional is a functional on the space of Kähler potentials of a compact Kähler manifold whose critical points are constant scalar curvature Kähler metrics. The Mabuchi functional was introduced by Toshiki Mabuchi in 1985 as a functional which integrates the Futaki invariant, which is an obstruction to the existence of a Kähler–Einstein metric on a Fano manifold.
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1118840802
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dbr:Moment_map dbr:Futaki_invariant dbr:Functional_(mathematics) dbr:Toshiki_Mabuchi dbr:Complex_geometry dbr:Ddbar_lemma dbc:Differential_geometry dbr:Symplectic_reduction dbr:Complex_manifold dbr:Kähler–Einstein_metric dbr:Contractible_space dbr:Kähler_manifold dbr:Mathematics dbr:Exact_differential_form dbr:Constant_scalar_curvature_Kähler_metric dbr:Kähler_potential dbr:Generalized_Stokes_theorem dbr:Geometric_invariant_theory dbr:De_Rham_cohomology dbr:Positive_form dbr:One-form dbr:Geodesic dbr:Donaldson–Futaki_invariant dbr:Scalar_curvature dbr:K-stability dbr:Riemannian_metric dbr:K-stability_of_Fano_varieties dbr:Cohomology_class dbr:Test_configuration dbr:Critical_point_(mathematics)
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dbo:abstract
In mathematics, and especially complex geometry, the Mabuchi functional or K-energy functional is a functional on the space of Kähler potentials of a compact Kähler manifold whose critical points are constant scalar curvature Kähler metrics. The Mabuchi functional was introduced by Toshiki Mabuchi in 1985 as a functional which integrates the Futaki invariant, which is an obstruction to the existence of a Kähler–Einstein metric on a Fano manifold. The Mabuchi functional is an analogy of the log-norm functional of the moment map in geometric invariant theory and symplectic reduction. The Mabuchi functional appears in the theory of K-stability as an analytical functional which characterises the existence of constant scalar curvature Kähler metrics. The slope at infinity of the Mabuchi functional along any geodesic ray in the space of Kähler potentials is given by the Donaldson–Futaki invariant of a corresponding test configuration. Due to the variational techniques of Berman–Boucksom–Jonsson in the study of Kähler–Einstein metrics on Fano varieties, the Mabuchi functional and various generalisations of it have become critically important in the study of K-stability of Fano varieties, particularly in settings with singularities.
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