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

Guoyuan Chen; Yong Liu; Juncheng Wei

doi:10.3934/dcds.2019228

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2020, Volume 40, Issue 6: 3215-3233. Doi: 10.3934/dcds.2019228 open access

This issue Previous Article On gaussian curvature equation in $ \mathbb{R}^2 $ with prescribed nonpositive curvature Next Article A Hopf lemma and regularity for fractional $ p $-Laplacians

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

1.
School of Data Sciences, Zhejiang University of Finance & Economics, Hangzhou 310018, Zhejiang, China
2.
School of Mathematics, University of Science and Technology of China, Hefei, China
3.
Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada, V6T 1Z2

^* Corresponding author: Juncheng Wei
^* Corresponding author: Juncheng Wei

Dedicated to Professor Wei-Ming Ni on the occasion of his 70th birthday

Received: August 2018

Revised: February 2019

Published: March 2020

The first author is supported by ZPNFSC (No. LY18A010023), the third author is partially supported by NSERC of Canada

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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 those harmonic maps near $ u $ and hence the real dimension of bounded kernel space of $ L_u $ is $ 4|m|+2 $.

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Mathematics Subject Classification: Primary: 53C43; Secondary: 35J60.

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