On p-harmonic self-maps of spheres

TL;DR

This paper investigates p-harmonic self-maps of spheres, establishing the existence of infinitely many such maps for certain dimensions and analyzing their stability, especially for the identity map.

Contribution

It proves the existence of infinitely many p-harmonic self-maps of spheres within specific dimension ranges and explicitly determines the spectrum of the Jacobi operator for the identity map.

Findings

Existence of infinitely many p-harmonic self-maps for p<m<2+p+2√p.

Explicit spectrum of the Jacobi operator for the identity map.

Stability of the identity map when p>m.

Abstract

In this manuscript we study rotationally $p$-harmonic maps between spheres. We prove that for $p\in\mathbb{N}$ given, there exist infinitely many $p$-harmonic self-maps of $\mathbb{S}^m$ for each $m\in\mathbb{N}$ with $p<m< 2+p+2\sqrt{p}$. In the case of the identity map of $\mathbb{S}^m$ we explicitly determine the spectrum of the corresponding Jacobi operator and show that for $p>m$, the identity map of $\mathbb{S}^m$ is stable when interpreted as a $p$-harmonic self-map of $\mathbb{S}^m$.