June  2020, 40(6): 3215-3233. 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, 2020, 40 (6) : 3215-3233. doi: 10.3934/dcds.2019228
References:
[1]

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

[2]

F. CohenR. CohenB. 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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8] M. A. Guest, Harmonic Maps, Loop Groups, and Integrable Systems, Vol. 38, Cambridge University Press, 1997.  doi: 10.1017/CBO9781139174848.  Google Scholar
[9]

S. GustafsonK. 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

[10]

S. GustafsonK. 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

[11]

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

[12]

J. Jost, Harmonic maps between surfaces: (with a special chapter on conformal mappings), Vol. 1062, Springer, 2006. doi: 10.1007/BFb0100160.  Google Scholar

[13]

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

[14]

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

[15]

F. Lin and C. Wang, The Analysis of Harmonic Maps and Their Heat Flows, World Scientific, 2008. doi: 10.1142/9789812779533.  Google Scholar

[16]

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

[17]

E. Outerelo and J. M. Ruiz, Mapping Degree Theory, Vol. 108, American Mathematical Soc., 2009. doi: 10.1090/gsm/108.  Google Scholar

[18]

R. M. Schoen and S.-T. Yau, Lectures on Harmonic Maps, Vol. 2, Amer. Mathematical Society, 1997.  Google Scholar

[19]

G. Segal, The topology of spaces of rational functions, Acta Mathematica, 143 (1979), 39-72.  doi: 10.1007/BF02392088.  Google Scholar

[20]

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

[21]

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

[22]

J. WeiC. 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

show all references

References:
[1]

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

[2]

F. CohenR. CohenB. 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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8] M. A. Guest, Harmonic Maps, Loop Groups, and Integrable Systems, Vol. 38, Cambridge University Press, 1997.  doi: 10.1017/CBO9781139174848.  Google Scholar
[9]

S. GustafsonK. 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

[10]

S. GustafsonK. 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

[11]

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

[12]

J. Jost, Harmonic maps between surfaces: (with a special chapter on conformal mappings), Vol. 1062, Springer, 2006. doi: 10.1007/BFb0100160.  Google Scholar

[13]

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

[14]

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

[15]

F. Lin and C. Wang, The Analysis of Harmonic Maps and Their Heat Flows, World Scientific, 2008. doi: 10.1142/9789812779533.  Google Scholar

[16]

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

[17]

E. Outerelo and J. M. Ruiz, Mapping Degree Theory, Vol. 108, American Mathematical Soc., 2009. doi: 10.1090/gsm/108.  Google Scholar

[18]

R. M. Schoen and S.-T. Yau, Lectures on Harmonic Maps, Vol. 2, Amer. Mathematical Society, 1997.  Google Scholar

[19]

G. Segal, The topology of spaces of rational functions, Acta Mathematica, 143 (1979), 39-72.  doi: 10.1007/BF02392088.  Google Scholar

[20]

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

[21]

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

[22]

J. WeiC. 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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