Bergman metric
E521039
The Bergman metric is a canonical Kähler metric on complex domains derived from the Bergman kernel, widely used in several complex variables and complex differential geometry.
Statements (42)
| Predicate | Object |
|---|---|
| instanceOf |
Hermitian metric
ⓘ
Kähler metric ⓘ canonical metric ⓘ |
| appliesTo |
bounded symmetric domains
ⓘ
pseudoconvex domains ⓘ |
| category | complex Finsler and Riemannian-type metrics ⓘ |
| constructedFrom | Bergman kernel NERFINISHED ⓘ |
| curvatureProperty |
has constant holomorphic sectional curvature on the unit ball
ⓘ
has negative holomorphic sectional curvature on the unit ball ⓘ |
| definedOn |
bounded domains in ℂⁿ
ⓘ
complex domains ⓘ |
| definedVia |
Levi form of the logarithm of the Bergman kernel
ⓘ
second derivatives of log K(z,z) where K is the Bergman kernel ⓘ |
| dependsOn | space of square-integrable holomorphic functions ⓘ |
| determines |
an intrinsic complex structure compatible Riemannian metric
ⓘ
the Bergman distance NERFINISHED ⓘ |
| field |
complex differential geometry
ⓘ
several complex variables ⓘ |
| gives | a canonical volume form on the domain ⓘ |
| introducedBy | Stefan Bergman NERFINISHED ⓘ |
| introducedIn | 20th century ⓘ |
| invariantUnder |
automorphism group of the domain
ⓘ
biholomorphic maps ⓘ |
| is |
Kähler-Einstein on the unit ball in ℂⁿ
ⓘ
complete on bounded homogeneous domains ⓘ invariant under biholomorphic automorphisms of the domain ⓘ real-analytic on the domain ⓘ unique up to biholomorphic equivalence for a given domain ⓘ |
| regularity | smooth on strongly pseudoconvex domains ⓘ |
| relatedTo |
Carathéodory metric
NERFINISHED
ⓘ
Kobayashi metric NERFINISHED ⓘ Poincaré metric NERFINISHED ⓘ |
| restrictionProperty | restricts to the Poincaré metric on the unit disc ⓘ |
| usedFor |
biholomorphic classification of domains
ⓘ
complex Monge–Ampère equations ⓘ defining invariant distances on complex domains ⓘ studying automorphism groups of domains ⓘ studying intrinsic geometry of complex domains ⓘ |
| usedIn |
complex algebraic geometry
ⓘ
complex analysis ⓘ geometric function theory ⓘ theory of several complex variables ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.