Totally geodesic discs in bounded symmetric domains - Kim, Sung-Yeon; Seo, Aeryeong (Complex analysis and its synergies, 2022)
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.