Calabi conjecture

E888043

mathematical conjecture result in complex differential geometry

The Calabi conjecture is a fundamental result in complex differential geometry, proved by Shing-Tung Yau, which characterizes when a compact Kähler manifold admits a unique Ricci-flat Kähler metric in a given Kähler class.

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All labels observed (1)

Label	Occurrences
Calabi conjecture canonical	2

Statements (48)

Predicate	Object
instanceOf	mathematical conjecture ⓘ result in complex differential geometry ⓘ
concerns	Calabi–Yau manifolds NERFINISHED ⓘ Kähler classes NERFINISHED ⓘ Ricci curvature ⓘ Ricci-flat Kähler metrics ⓘ compact Kähler manifolds ⓘ complex Monge–Ampère equation NERFINISHED ⓘ first Chern class ⓘ
dimension	holds in all complex dimensions ⓘ
field	Kähler geometry NERFINISHED ⓘ Riemannian geometry ⓘ algebraic geometry ⓘ complex differential geometry ⓘ
formulatedBy	Eugenio Calabi NERFINISHED ⓘ
generalizationOf	problems of finding metrics with prescribed Ricci curvature ⓘ
hasConsequence	applications in string theory via Calabi–Yau compactifications ⓘ classification of Calabi–Yau manifolds as Ricci-flat Kähler manifolds with vanishing first Chern class ⓘ construction of metrics with prescribed Ricci form ⓘ existence of Kähler–Einstein metrics with zero Ricci curvature ⓘ
implies	existence of Calabi–Yau metrics ⓘ existence of Ricci-flat metrics on K3 surfaces ⓘ existence of Ricci-flat metrics on complex tori ⓘ
influenced	development of Calabi–Yau geometry ⓘ research in string theory compactifications ⓘ study of Kähler–Einstein metrics ⓘ
namedAfter	Eugenio Calabi NERFINISHED ⓘ
originallyFormulated	1950s GENERATED ⓘ
provedBy	Shing-Tung Yau NERFINISHED ⓘ
provedUsing	Moser iteration NERFINISHED ⓘ Schauder estimates NERFINISHED ⓘ a priori estimates ⓘ continuity method ⓘ maximum principle ⓘ
relatedTo	Aubin–Yau theorem NERFINISHED ⓘ Calabi–Yau manifold NERFINISHED ⓘ Kähler–Einstein metric NERFINISHED ⓘ Yau's theorem NERFINISHED ⓘ
requiresCondition	compactness of the Kähler manifold ⓘ fixed Kähler class ⓘ prescribed first Chern class ⓘ
states	that on a compact Kähler manifold with vanishing first Chern class there exists a Ricci-flat Kähler metric in any given Kähler class ⓘ that the Ricci-flat Kähler metric in a fixed Kähler class is unique ⓘ
status	proved ⓘ
uses	complex Monge–Ampère equation NERFINISHED ⓘ nonlinear elliptic partial differential equations ⓘ
yearProved	1976 ⓘ 1977 ⓘ

Referenced by (2)

Full triples — surface form annotated when it differs from this entity's canonical label.

Kähler–Ricci flow → relatedTo → Calabi conjecture ⓘ

Monge–Ampère equation → usedIn → Calabi conjecture ⓘ