Totally geodesic discs in bounded symmetric domains

In this paper, we characterize C 2 -smooth totally geodesic isometric embeddings f : Ω → Ω ′ between bounded symmetric domains Ω and Ω ′ which extend C 1 -smoothly over some open subset in the Shilov boundaries and have nontrivial normal derivatives on it. In particular, if Ω is irreducible, there e...

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Bibliographic Details
Published in:Complex analysis and its synergies Vol. 8; no. 3
Main Authors: Kim, Sung-Yeon, Seo, Aeryeong
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.09.2022
Springer Nature B.V
Subjects:
Algebraic Geometry
Analysis
Domains
Dynamical Systems and Ergodic Theory
Functional Analysis
Functions of a Complex Variable
Mappings and symmetry in complex and CR geometry
Mathematical analysis
Mathematics
Mathematics and Statistics
Partial Differential Equations
32M15
53C35
Totally geodesic isometric embedding
53C55
Bounded symmetric domain
Holomorphicity
Bergman metric
ISSN:2524-7581, 2197-120X
Online Access:Get full text
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Summary:In this paper, we characterize C 2 -smooth totally geodesic isometric embeddings f : Ω → Ω ′ between bounded symmetric domains Ω and Ω ′ which extend C 1 -smoothly over some open subset in the Shilov boundaries and have nontrivial normal derivatives on it. In particular, if Ω is irreducible, there exist totally geodesic bounded symmetric subdomains Ω 1 and Ω 2 of Ω ′ such that f = ( f 1 , f 2 ) maps into Ω 1 × Ω 2 ⊂ Ω where f 1 is holomorphic and f 2 is anti-holomorphic totally geodesic isometric embeddings. If rank ( Ω ′ ) < 2 rank ( Ω ) , then either f or f ¯ is a standard holomorphic embedding.
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ISSN:2524-7581
2197-120X
DOI:10.1007/s40627-022-00098-z