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

TL;DR

This paper proves that the standard half-harmonic map from the real line to the circle is nondegenerate, meaning all bounded solutions to the linearized equation are generated by symmetries like dilation, translation, and rotation.

Contribution

It establishes the nondegeneracy of the standard half-harmonic map into the circle, a key property for understanding its stability and uniqueness.

Findings

All bounded solutions of the linearized equation are symmetry-generated.

The half-harmonic map is nondegenerate in the sense of linear stability.

The result characterizes the kernel of the linearized operator.

Abstract

We prove that the standard half-harmonic map $U:\mathbb{R}\to\mathbb{S}^1$ defined by \begin{equation*} x\to \begin{pmatrix} \frac{x^2-1}{x^2+1} \frac{-2x}{x^2+1} \end{pmatrix} \end{equation*} 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 $U$.