On the equivariant stability of harmonic self-maps of cohomogeneity one manifolds

TL;DR

This paper studies the stability of harmonic self-maps on cohomogeneity one manifolds, providing explicit solutions to the Jacobi equation and demonstrating stability in specific cases like spheres and special orthogonal groups.

Contribution

It offers the first explicit solutions to the Jacobi equation for harmonic self-maps in cohomogeneity one settings and analyzes their equivariant stability.

Findings

Explicit solutions to the Jacobi equation for certain harmonic self-maps.

Identification of cases where the identity map is equivariantly stable.

Analysis of harmonic self-maps on spheres, SO(n), and SU(3).

Abstract

The systematic study of harmonic self-maps on cohomogeneity one manifolds has recently been initiated by P\"uttmann and the second named author in \cite{MR4000241}. In this article we investigate the corresponding Jacobi equation describing the equivariant stability of such harmonic self-maps. Besides several general statements concerning their equivariant stability we explicitly solve the Jacobi equation for some harmonic self-maps in the cases of spheres, special orthogonal groups and $\SU(3)$. In particular, we show by an explicit calculation that for specific cohomogeneity one actions on the sphere the identity map is equivariantly stable.