Construction of harmonic maps between cohomogeneity one manifolds

TL;DR

The paper studies equivariant harmonic maps between cohomogeneity-one manifolds by reducing the harmonic-map equations to one-dimensional ODEs along a normal geodesic and exploiting symmetry to obtain existence results. It develops two main constructions: first, harmonic self-maps of round spheres for dimensions $n\in\{3,4,5,7\}$ via symmetric boundary-value problems and metric deformations, supported by a Lyapunov-type function and numerical evidence; second, non-compact cohomogeneity-one manifolds where metric deformations yield non-trivial equivariant harmonic maps between $(M,\tilde{g})$ and $(M,g)$ (rendering-type results). The key contributions include proving the existence of four harmonic self-maps in specific spheres and establishing a general deformation-based existence theorem (Theorem B) for harmonic maps in the non-compact setting, highlighting the utility of symmetry reductions and boundary-value analyses in geometric analysis. Together, these results advance understanding of how metric deformations interact with symmetry to generate harmonic maps in both compact and non-compact cohomogeneity-one contexts, with potential implications for geometric flow and rendering-type problems.$

Abstract

We construct equivariant harmonic maps between cohomogeneity one manifolds.

Theorems & Definitions (39)

• Theorem 1.1
• Theorem 1.2
• Theorem 2.1: MR4400726
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Lemma 2.5
• Theorem 2.6: Theorem E in MR4000241
• Theorem 2.7: see Theorem H in MR4000241
• Lemma 3.1: Properties of $h_1$