Nondegeneracy of harmonic maps from $\mathbb R^2$ to $\mathbb S^2$

TL;DR

This paper proves that all finite energy harmonic maps from the plane to the sphere are nondegenerate, meaning their linearized operators have a kernel structure precisely characterized by the maps themselves, with a specific dimension.

Contribution

It establishes the nondegeneracy of harmonic maps from  to , providing a detailed description of the kernel of the linearized operator at these maps.

Findings

All finite energy harmonic maps are nondegenerate.

Kernel dimension of the linearized operator is 4|m|+2.

Kernel maps are generated by harmonic maps near each solution.

Abstract

We prove that all harmonic maps from $\mathbb R^2$ to $\mathbb S^2$ with finite energy are nondegenerate. That is, for any harmonic map $u$ from $\mathbb R^2$ to $\mathbb S^2$ of degree $m$ (in $\mathbb Z$), all bounded kernel maps of the linearized operator $L_u$ at $u$ are generated by these harmonic maps near $u$ and hence the real dimension of bounded kernel space of $L_u$ is $4|m|+2$.