Day-Taheri2020_Article_StabilityAndLocalMinimalityOfS.pdf (4.81 MB)
Stability and local minimality of spherical harmonic twists u=Q(|x|)x|x|-1, positivity of second variations and conjugate points on SO(n)
Version 2 2023-06-12, 09:05
Version 1 2023-06-09, 17:45
journal contribution
posted on 2023-06-12, 09:05 authored by George Morrison, Ali TaheriAli TaheriIn 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.
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Publication status
- Published
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- Published version
Journal
The Journal of AnalysisISSN
0971-3611Publisher
SpringerExternal DOI
Issue
2Volume
28Page range
431-460Department affiliated with
- Mathematics Publications
Research groups affiliated with
- Analysis and Partial Differential Equations Research Group Publications
Full text available
- Yes
Peer reviewed?
- Yes
Legacy Posted Date
2019-05-09First Open Access (FOA) Date
2019-06-06First Compliant Deposit (FCD) Date
2019-05-09Usage metrics
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