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

Abstract

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;Xn,Sn-1]:=Δu+|∇u|2u=0inXn,|u|=1inXn,u=φon∂Xn,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\mathbf{HME}[u;\\, \\mathbb {X}^n, \\mathbb {S}^{n-1}] :=\\left\\{ \\begin{array}{ll} \\Delta u + |\\nabla u|^2 u =0 &{} \\qquad \\text { in } \\mathbb {X}^n, \\\\ |u|=1 &{}\\qquad \\text { in } \\mathbb {X}^n , \\\\ u = \\varphi &{}\\qquad \\text { on } \\partial \\mathbb {X}^n, \\end{array}\\right. \\end{aligned}$$\\end{document}by way of examining the positivityof the second variation of the associated Dirichlet energy. Here, following [31], by a spherical twist we mean a map u in mathscr {W}^{1,2}(mathbb {X}^n, mathbb {S}^{n-1}) of the form x mapsto {mathbf {Q}}(|x|)x|x|^{-1} where {mathbf {Q}}={mathbf {Q}}(r) lies in mathscr {C}([a, b], mathbf{SO}(n)) and {mathbb {X}}^n={x in mathbb {R}^n : a<|x|<b} (nge 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 {mathbf {Q}}(r)=mathrm{exp}(mathscr {G}(r) {mathbf {H}}) and geodesics on mathbf{SO}(n) is undertaken and its remarkable implication and interplay on the minimalityof spherical harmonic twists exploited.