Minimal Lie group homomorphisms

2009/08/09 by A. Caminha, Caminha, A.
Mathematics · Physics and Astronomy · #53C30 #53C42 #53C43 #Advanced Differential Geometry Research #Combinatorics #Differential Geometry (math.DG) #FOS: Mathematics #Fundamental representation #Geometric Analysis and Curvature Flows #Geometry #Geometry and complex manifolds #Homomorphism #Invariant (physics) #Lie algebra #Lie group #Mathematical physics #Mathematics #Maximal torus #Pure mathematics #Simple Lie group #Torus #math.DG #msc:53C30 #msc:53C42 #msc:53C43

paper · pdf · doi:10.48550/arxiv.0908.1259

This paper has been withdraw due to a crucial sign error in the proof of the theorem

openalex publication_date 2009/08/09 · arxiv created 2012/09/27 · arxiv updated 2012/09/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

Let G₁ and G₂ be Lie groups furnished with bi-invariant metrics and f:G₁→ G₂ be a Lie group homomorphism which is also a minimal isometric immersion. If G₁ is compact and connected, we prove that either G₁ 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.

Citations

• Harmonic Mappings of Riemannian Manifolds

Related

• Existence of isometric immersions into nilpotent Lie groups
• From homogeneous metric spaces to Lie groups
• Unstable minimal surfaces in symmetric spaces of non-compact type
• Riemannian submersions from simple, compact Lie groups
• Harmonic maps into sub-Riemannian Lie groups