    doi: 10.3934/dcds.2019228

## 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

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

Received  August 2018 Revised  February 2019 Published  June 2019

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

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 $4|m|+2$.

Citation: Guoyuan Chen, Yong Liu, Juncheng Wei. Nondegeneracy of harmonic maps from ${{\mathbb{R}}^{2}}$ to ${{\mathbb{S}}^{2}}$. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2019228
##### References:
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##### References:
  S. Chanillo and A. Malchiodi, Asymptotic morse theory for the equation ${\Delta} v = 2v_x\wedge v_y$, Comm. Anal. Geom, 13 (2005), 187-251. doi: 10.4310/CAG.2005.v13.n1.a6.  Google Scholar  F. Cohen, R. Cohen, B. Mann and R. Milgram, The topology of rational functions and divisors of surfaces, Acta Mathematica, 166 (1991), 163-221. doi: 10.1007/BF02398886.  Google Scholar  J. Davila, M. del Pino and J. Wei, Singularity formation for the two-dimensional harmonic map flow into $\mathbb{S}^2$, preprint, arXiv: 1702.05801.Google Scholar  J. Eells and L. Lemaire, A report on harmonic maps, Bulletin of the London Mathematical Society, 10 (1978), 1-68. doi: 10.1112/blms/10.1.1.  Google Scholar  J. Eells and L. Lemaire, Another report on harmonic maps, Bulletin of the London Mathematical Society, 20 (1988), 385-524. doi: 10.1112/blms/20.5.385.  Google Scholar  J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, American Journal of Mathematics, 86 (1964), 109-160. doi: 10.2307/2373037.  Google Scholar  J. Eells and C. Wood, Harmonic maps from surfaces to complex projective spaces, Advances in Mathematics, 49 (1983), 217-263. doi: 10.1016/0001-8708(83)90062-2.  Google Scholar  M. A. Guest, Harmonic Maps, Loop Groups, and Integrable Systems, Vol. 38, Cambridge University Press, 1997. doi: 10.1017/CBO9781139174848.  Google Scholar  S. Gustafson, K. Kang and T.-P. Tsai, Schrödinger flow near harmonic maps, Communications on Pure Applied Mathematics, 60 (2007), 463-499. doi: 10.1002/cpa.20143.  Google Scholar  S. Gustafson, K. Kang and T.-P. Tsai, Asymptotic stability of harmonic maps under the Schrödinger flow, Duke Mathematical Journal, 145 (2008), 537-583. doi: 10.1215/00127094-2008-058.  Google Scholar  F. Hélein and J. C. Wood, Harmonic maps, Handbook of global analysis, 1213 (2008), 417-491. doi: 10.1016/B978-044452833-9.50009-7.  Google Scholar  J. Jost, Harmonic maps between surfaces: (with a special chapter on conformal mappings), Vol. 1062, Springer, 2006. doi: 10.1007/BFb0100160.  Google Scholar  E. Lenzmann and A. Schikorra, On energy-critical half-wave maps into $\mathbb{S}^2$, Inventiones Mathematicae, 1–82. doi: 10.1007/s00222-018-0785-1.  Google Scholar  C.-S. Lin, J. Wei and D. Ye, Classification and nondegeneracy of $S{U}(n+1)$ Toda system with singular sources, Inventiones Mathematicae, 190 (2012), 169–207. doi: 10.1007/s00222-012-0378-3.  Google Scholar  F. Lin and C. Wang, The Analysis of Harmonic Maps and Their Heat Flows, World Scientific, 2008. doi: 10.1142/9789812779533.  Google Scholar  F. Merle, P. Raphaël and I. Rodnianski, Blowup dynamics for smooth data equivariant solutions to the critical Schrödinger map problem, Inventiones Mathematicae, 193 (2013), 249–365. doi: 10.1007/s00222-012-0427-y.  Google Scholar  E. Outerelo and J. M. Ruiz, Mapping Degree Theory, Vol. 108, American Mathematical Soc., 2009. doi: 10.1090/gsm/108.  Google Scholar  R. M. Schoen and S.-T. Yau, Lectures on Harmonic Maps, Vol. 2, Amer. Mathematical Society, 1997. Google Scholar  G. Segal, The topology of spaces of rational functions, Acta Mathematica, 143 (1979), 39-72. doi: 10.1007/BF02392088.  Google Scholar  Y. Sire, J. Wei and Y. Zheng, Nondegeneracy of half-harmonic maps from $\mathbb{R}$ into $\mathbb{S}^1$, preprint, arXiv: 1701.03629. doi: 10.1090/proc/14184.  Google Scholar  K. Uhlenbeck, Harmonic maps into Lie groups: classical solutions of the chiral model, Journal of Differential Geometry, 30 (1989), 1-50. doi: 10.4310/jdg/1214443286.  Google Scholar  J. Wei, C. Zhao and F. Zhou, On nondegeneracy of solutions to $SU(3)$ Toda system, Comptes Rendus Mathematique, 349 (2011), 185-190. doi: 10.1016/j.crma.2010.11.025.  Google Scholar
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