About: Mabuchi functional

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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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rdfs:label	Mabuchi functional (en)
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. (en)
dcterms:subject	Differential geometry
Wikipage page ID	68495772 (xsd:integer)
Wikipage revision ID	1118840802 (xsd:integer)
Link from a Wikipage to another Wikipage	Scalar curvature Ddbar lemma De Rham cohomology Kähler–Einstein metric Positive form Differential geometry Critical point (mathematics) Mathematics Functional (mathematics) Generalized Stokes theorem Geodesic Geometric invariant theory Constant scalar curvature Kähler metric Contractible space Symplectic reduction Complex geometry Complex manifold Toshiki Mabuchi K-stability K-stability of Fano varieties Riemannian metric Donaldson–Futaki invariant Kähler manifold Linear form Moment map Kähler potential Test configuration Futaki invariant Cohomology class Exact differential form
sameAs	Mabuchi functional Mabuchi functional
dbp:wikiPageUsesTemplate	dbt:Reflist
has 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. (en)
prov:wasDerivedFrom	wikipedia-en:Mabuchi_functional?oldid=1118840802&ns=0
page length (characters) of wiki page	5659 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Mabuchi_functional
is Link from a Wikipage to another Wikipage of	Constant scalar curvature Kähler metric Toshiki Mabuchi K-energy K-energy functional
is Wikipage redirect of	K-energy K-energy functional
is foaf:primaryTopic of	wikipedia-en:Mabuchi_functional

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