Spherical Twists as the σ₂-Harmonic Maps from n-Dimensional Annuli into $\mathbb {S}^{n-1}$

Abstract

Let $\mathbb {X} \subset \mathbb {R}^{n}$ be a bounded Lipschitz domain and consider the σ₂-energy functional

$${\mathbb{F}_{\sigma_{2}}}[u; \mathbb{X}] := {\int}_{\mathbb{X}} \left|\wedge^{2} \nabla u\right|^{2} \, dx, $$

over the space of admissible Sobolev maps

$$\mathcal{A}(\mathbb{X}) :=\left\{u \in W^{1,4}(\mathbb{X}, \mathbb{S}^{n-1}) : u|_{\partial \mathbb{X}} = x|x|^{-1}\right\}. $$

In this article we address the question of multiplicity versus uniqueness for extremals and strong local minimisers of the σ₂-energy funcional $\mathbb F_{\sigma _{2}}[\cdot , \mathbb {X}]$ in $\mathcal {A}(\mathbb {X})$ where the domain $\mathbb X$ is n-dimensional annuli. We consider a topological class of maps referred to as spherical twists and examine them in connection with the Euler-Lagrange equations associated with σ₂-energy functional over $\mathcal {A}(\mathbb {X})$, the so-called σ₂-harmonic map equation on $\mathbb {X}$. The main result is a surprising discrepancy between even and odd dimensions. In even dimensions the latter system of equations admits infinitely many smooth solutions amongst such maps whereas in odd dimensions this number reduces to one. The result relies on a careful analysis of the full versus the restricted Euler-Lagrange equations.