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On proper holomorphic mappings among irreducible bounded symmetric domains of rank at least $2$


Author: Sui-Chung Ng
Journal: Proc. Amer. Math. Soc. 143 (2015), 219-225
MSC (2010): Primary 32H35, 32M15
DOI: https://doi.org/10.1090/S0002-9939-2014-12226-X
Published electronically: September 15, 2014
MathSciNet review: 3272747
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Abstract: We give a characterization for totally geodesic embeddings between higher-rank irreducible bounded symmetric domains in terms of certain totally geodesic rank-1 symmetric subspaces. More explicitly, we prove that for two irreducible bounded symmetric domains $\Omega _1,\Omega _2$ of rank at least 2, a holomorphic map $F:\Omega _1\rightarrow \Omega _2$ is a totally geodesic embedding with respect to the Bergman metrics if $F$ maps the minimal disks of $\Omega _1$ into rank-1 characteristic symmetric subspaces of $\Omega _2$. As a simple corollary, we obtain a much simpler proof for a theorem of Tsai which says that $F$ is totally geodesic if $F$ is proper and $\operatorname {rank}(\Omega _1) = \operatorname {rank}(\Omega _2)$.


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Sui-Chung Ng
Affiliation: Department of Mathematics, Temple University, 1805 N. Broad Street, Philadelphia, Pennsylvania 19122
Email: scng@temple.edu

Received by editor(s): October 15, 2012
Received by editor(s) in revised form: March 20, 2013
Published electronically: September 15, 2014
Communicated by: Franc Forstneric
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.