
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 and be Lie groups furnished with bi-invariant metrics and be a Lie group homomorphism which is also a minimal isometric immersion. If is compact and connected, we prove that either is isometric to a flat torus or is unstable as a harmonic map. We also apply this result to the case in which 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.
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Geometry and complex manifolds · Advanced Differential Geometry Research
