Minimal Lie group homomorphisms

TL;DR

This paper investigates minimal isometric Lie group homomorphisms between Lie groups with bi-invariant metrics, establishing conditions under which such maps are either flat tori or unstable, with applications to subgroups and examples.

Contribution

It proves a classification result for minimal isometric homomorphisms from compact, connected Lie groups, showing they are either flat tori or unstable as harmonic maps.

Findings

Either $G_1$ is a flat torus or $f$ is unstable as a harmonic map.

Application to subgroups of Lie groups.

Construction of unstable harmonic maps into the orthogonal group.

Abstract

Let $G_1$ and $G_2$ be Lie groups furnished with bi-invariant metrics and $f:G_1\rightarrow G_2$ be a Lie group homomorphism which is also a minimal isometric immersion. If $G_1$ is compact and connected, we prove that either $G_1$ is isometric to a flat torus or $f$ is unstable as a harmonic map. We also apply this result to the case in which $f$ is the inclusion of a compact, connected Lie subgroup of a Lie group, as well as to construct several examples of unstable harmonic maps into the orthogonal group.