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Nondegeneracy of half-harmonic maps from $ \mathbb{R}$ into $ \mathbb{S}^1$

Authors: Yannick Sire, Juncheng Wei and Youquan Zheng
Journal: Proc. Amer. Math. Soc. 146 (2018), 5263-5268
MSC (2010): Primary 35B06
Published electronically: September 17, 2018
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Abstract: We prove that the standard half-harmonic map $ U:\mathbb{R}\to \mathbb{S}^1$ defined by

$\displaystyle x\to \begin {pmatrix}\frac {x^2-1}{x^2+1} \\ [4pt] \frac {-2x}{x^2+1} \end{pmatrix}$    

is nondegenerate in the sense that all bounded solutions of the linearized half-harmonic map equation are linear combinations of three functions corresponding to rigid motions (dilation, translation, and rotation) of $ U$.

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Additional Information

Yannick Sire
Affiliation: Department of Mathematics, Johns Hopkins University, 404 Krieger Hall, 3400 N. Charles Street, Baltimore, Maryland 21218

Juncheng Wei
Affiliation: Department of Mathematics, University of British Columbia, Vancouver V6T 1Z2, Canada

Youquan Zheng
Affiliation: School of Mathematics, Tianjin University, Tianjin 300072, People’s Republic of China

Received by editor(s): December 20, 2017
Received by editor(s) in revised form: April 5, 2018
Published electronically: September 17, 2018
Additional Notes: The third author was partially supported by NSF of China (11301374) and China Scholarship Council (CSC)
Communicated by: Joachim Krieger
Article copyright: © Copyright 2018 American Mathematical Society

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