Lempert function on convex domains

E521040

complex-analytic invariant distance-like function holomorphic invariant

The Lempert function on convex domains is a complex-analytic invariant that coincides with the Kobayashi distance and provides an extremal characterization of holomorphic mappings between convex domains in several complex variables.

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Statements (46)

Predicate	Object
instanceOf	complex-analytic invariant ⓘ distance-like function ⓘ holomorphic invariant ⓘ
appearsIn	Lempert’s 1981–1982 work on complex geodesics in convex domains ⓘ
appliesTo	bounded convex domains ⓘ unbounded convex domains ⓘ
associatedWith	extremal discs ⓘ
characterizes	extremal holomorphic mappings between convex domains ⓘ
coincidesWith	Kobayashi distance on convex domains ⓘ integrated Kobayashi distance on convex domains ⓘ
definedOn	convex domains in C^n ⓘ domains in several complex variables ⓘ
definedVia	extremal holomorphic discs ⓘ infimum over holomorphic discs joining two points ⓘ
dependsOn	complex structure of the domain ⓘ
generalizes	Poincaré distance on the unit disc ⓘ
hasProperty	biholomorphic invariance ⓘ monotonicity with respect to domain inclusion ⓘ
hasSpecialCase	Lempert function on convex balanced domains ⓘ Lempert function on strictly convex domains ⓘ Lempert function on the unit ball in C^n ⓘ
introducedBy	László Lempert NERFINISHED ⓘ
invariantUnder	biholomorphic mappings ⓘ holomorphic automorphisms of the domain ⓘ
is	holomorphically contractible invariant ⓘ
isToolFor	studying biholomorphic equivalence of convex domains ⓘ studying extremal problems for holomorphic maps ⓘ
playsRoleIn	equivalence of invariant metrics on convex domains ⓘ theory of complex geodesics in convex domains ⓘ
relatedTo	Carathéodory distance NERFINISHED ⓘ Kobayashi distance NERFINISHED ⓘ Schwarz–Pick type inequalities in several variables ⓘ complex geodesics ⓘ holomorphic retracts of convex domains ⓘ intrinsic pseudodistances ⓘ
satisfies	triangle inequality ⓘ
studiedIn	complex analysis in several variables ⓘ geometric function theory in higher dimensions ⓘ
takesValuesIn	[0,+∞] ⓘ
usedIn	complex geometry ⓘ invariant metrics on complex domains ⓘ several complex variables ⓘ
usedToCharacterize	complex geodesics in convex domains ⓘ
usedToProve	equivalence of Kobayashi and Carathéodory distances on certain convex domains ⓘ
usedToStudy	holomorphic mappings between convex domains ⓘ intrinsic geometry of convex domains ⓘ

Referenced by (1)

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

Carathéodory metric → isUpperBoundFor → Lempert function on convex domains ⓘ