Stability and local minimality of spherical harmonic twists u=Q(|x|)x|x|-1, positivity of second variations and conjugate points on SO(n)

In this paper we discuss the stability and local minimising properties of spherical twists that arise as solutions to the harmonic map equation HME[u; X n , S n-1 ] := ? ?? ?? ?u + |?u| 2 u = 0 in X n , |u| = 1 in X n , u = ? on ?X n , by way of examining the positivity of the second variation of the associated Dirichlet energy. Here, following [30], by a spherical twist we mean a map u ? W 1,2 (X n , S n-1 ) of the form x 7? Q(|x|)x|x| -1 where Q = Q(r) lies in C ([a, b], SO(n)) and X n = {x ? R n : a < |x| < b} (n = 2). It is shown that subject to a structural condition on the twist path the energy at the associated spherical twist solution to the system has a positive definite second variation and subsequently proven to furnish a strong local energy minimiser. A detailed study of Jacobi fields and conjugate points along the twist path Q(r) = exp(G (r)H) and geodesics on SO(n) is undertaken and its remarkable implication and interplay on the minimality of spherical harmonic twists exploited.