Stability and local minimality of spherical harmonic twists $$u={\mathbf {Q}}(|x|) x|x|^{-1}$$ , positivity of second variations and conjugate points on $$\mathbf{SO}(n)$$
Tóm tắt
In this paper we discuss the stability and local minimising properties of spherical twists that arise as solutions to the harmonic map equation $$\begin{aligned} \mathbf{HME}[u;\, \mathbb {X}^n, \mathbb {S}^{n-1}] :=\left\{ \begin{array}{ll} \Delta u + |\nabla u|^2 u =0 &{} \qquad \text { in } \mathbb {X}^n, \\ |u|=1 &{}\qquad \text { in } \mathbb {X}^n , \\ u = \varphi &{}\qquad \text { on } \partial \mathbb {X}^n, \end{array}\right. \end{aligned}$$by way of examining the positivity
of 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|
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