Totally geodesic discs in bounded symmetric domains

TL;DR

This paper characterizes smooth totally geodesic isometric embeddings between bounded symmetric domains, revealing their structure and conditions under which they are holomorphic, anti-holomorphic, or standard embeddings.

Contribution

It provides a detailed classification of smooth totally geodesic isometric embeddings, including their extension properties and the relationship between domain ranks.

Findings

Embeddings extend smoothly over parts of the boundary with nontrivial derivatives.

Existence of decompositions into holomorphic and anti-holomorphic components.

Conditions under which embeddings are standard holomorphic or anti-holomorphic.

Abstract

In this paper, we characterize $C^2$-smooth totally geodesic isometric embeddings $f\colon \Omega\to\Omega'$ between bounded symmetric domains $\Omega$ and $\Omega'$ 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 $\Omega'$ 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 $\text{rank}(\Omega')<2\text{rank}(\Omega)$, then either $f$ or $\bar f$ is a standard holomorphic embedding.