Totally geodesic discs in bounded symmetric domains

Abstract

In this paper, we characterize $C^2$-smooth totally geodesic isometric embeddings $f:\Omega \to {\Omega }^{\prime }$ between bounded symmetric domains $\Omega$ and ${\Omega }^{\prime }$ which extend $C^1$-smoothly over some open subset in the Shilov boundaries and have nontrivial normal derivatives on it. In particular, if $\Omega$ is irreducible, there exist totally geodesic bounded symmetric subdomains ${\Omega }_1$ and ${\Omega }_2$ of ${\Omega }^{\prime }$ such that $f=(f_1,f_2)$ maps into ${\Omega }_1\times {\Omega }_2\subset \Omega$ where $f_1$ is holomorphic and $f_2$ is anti-holomorphic totally geodesic isometric embeddings. If $\text{rank}({\Omega }^{\prime })<2\text{rank}(\Omega )$, then either $f$ or $\bar{f}$ is a standard holomorphic embedding.

10 pages