The space of convex domains in complex Euclidean space

In this mostly expository article, we describe some properties of the space of convex domains in complex Euclidean space (endowed with the local Hausdorff topology). In particular, we give careful proofs that the Kobayashi metric, the Bergman kernel/metric, and the Kähler–Einstein metric are all con...

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Bibliographic Details
Published in:The Journal of geometric analysis Vol. 30; no. 2; pp. 1312 - 1358
Main Authors: Gaussier, Hervé, Zimmer, Andrew
Format: Journal Article
Language:English
Published: New York Springer US 01.04.2020
Springer Nature B.V
Springer
Subjects:
Abstract Harmonic Analysis
an Important Tool in Several Complex Variables
Automorphisms
Closures
Convex and Discrete Geometry
Differential Geometry
Domains
Dynamical Systems and Ergodic Theory
Euclidean geometry
Euclidean space
Fourier Analysis
Geometry
Global Analysis and Analysis on Manifolds
Invariant Metrics
Mathematics
Mathematics and Statistics
Topology
Invariant metrics and pseudodistances
Convex domains
32F17
Primary 32F45
ISSN:1050-6926, 1559-002X
Online Access:Get full text
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Summary:In this mostly expository article, we describe some properties of the space of convex domains in complex Euclidean space (endowed with the local Hausdorff topology). In particular, we give careful proofs that the Kobayashi metric, the Bergman kernel/metric, and the Kähler–Einstein metric are all continuous on the space of convex domains. The group of affine automorphisms acts on this space and we also describe the orbit closures for some special classes of domains.
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ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-019-00346-5