Generalised twists as elastic energy extremals on annuli, Squaternions and lifting twist loops to the spinor groups

Let X = {x ? Rn : a < |x| < b} be a generalized annulus and consider the Dirichlet energy functional F[u; X] = 1/2?x |?u(x)|²dx, over the space of admissible maps A?(X) = {u ? W¹,²(X, Rn) : det ?u =1 a.e. in X, u|?X = ?}, where ? is the identity map. In this paper we consider a class of maps referred to as generalized twists and examine them in connection with the Euler–Lagrange equation associated with F[·, X] on A?(X). The approach is novel and is based on lifting twist loops from SO(n) to its double cover Spin(n) and reformulating the equations accordingly. We restrict our attention to low dimensions and prove that for n = 4 the system admits infinitely many smooth solutions in the form of twists while for n = 3 this number sharply reduces to one. We discuss some qualitative features of these solutions in view of their remarkable explicit representation through the exponential map of Spin(n).