Holomorphic maps between closed $SU(\ell, m)$-orbits in Grassmannian manifolds

In this paper, we study germs of smooth CR mappings sending a closed orbit of $SU(\ell, m)$ into a closed orbit of $SU(\ell^\prime , m^\prime)$ in Grassmannian manifolds. We show that if the signature difference of the Levi forms of two orbits is not too large, then the mapping can be factored into a simple form and one of the factors extends to a totally geodesic embedding of the ambient Grassmannian with respect to the standard metric. As an application, we give a sufficient condition for a proper holomorphic mapping between type I bounded symmeric domains to be the product of trivial embedding and a holomorphic mapping into a subdomain.

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This research was supported by Basic Science Research Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Science, ICT and Future Planning (grant number NRF-2015R1A2A2A11001367).

Received 3 June 2019

Accepted 11 December 2019

Published 2 June 2021