Spherical twists, stationary loops and harmonic maps from generalised annuli into spheres

Let X ? Rn be a generalised annulus and consider the Dirichlet energy functional E[u; X] := 1/2?X |?u(x)|²dx, on the space of admissible maps A?(X) = u ? W²,¹ (X, Sn¯¹) : u|?X = ? . Here ? ? C(?X, Sn¯¹) is fixed and A?(X) is non-empty. In this paper we introduce a class of maps referred to as spherical twists and examine them in connection with the Euler–Lagrange equation associated with E[·, X] on A?(X) [the so-called harmonic map equation on X]. The main result here is an interesting discrepancy between even and odd dimensions. Indeed for even n subject to a compatibility condition on ? the latter system admits infinitely many smooth solutions modulo isometries whereas for odd n this number reduces to one or none. We discuss qualitative features of the solutions in view of their novel and explicit representation through the exponential map of the compact Lie group SO(n).