Extrinsic polyharmonic maps into the sphere

Ali Fardoun, Stefano Montaldo, Cezar Oniciuc and Andrea Ratto

Vol. 331 (2024), No. 2, 259–281

DOI: 10.2140/pjm.2024.331.259

Abstract

In the first part we shall prove that the inverse of the stereographic projection ${\pi }^{-1}:{ℝ}^n\to {\mathbb{𝕊}}^n$ $(n\ge 2)$ is extrinsically $k$-harmonic if and only if $n=2k$. In  the second part we shall study minimizing properties and stability of its restriction to the closed ball $B^n(R)$. In this context we shall prove that there exists a small enough positive upper bound $R_k^{\ast }$ such that ${\pi }^{-1}:B^n(R)\to {\mathbb{𝕊}}^n$ is a minimizer provided that $0<R\le R_k^{\ast }$. By contrast, we shall show that ${\pi }^{-1}:B^n(R)\to {\mathbb{𝕊}}^n$ is not energy minimizing when $R>1$. Finally, in some cases we shall obtain stability with respect to rotationally symmetric variations (equivariant stability) for values of $R$ which are greater than $1$.

Keywords

polyharmonic maps, energy minimizing maps, stability, stereographic projection, conformal maps

Mathematical Subject Classification

Primary: 58E20

Secondary: 35J48

Milestones

Received: 21 February 2024

Accepted: 21 September 2024

Published: 30 October 2024