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Harmonic Self-Maps of \(\mathrm {SU}(3)\)

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Abstract

By constructing solutions of a singular boundary value problem, we prove the existence of a countably infinite family of harmonic self-maps of \(\mathrm {SU}(3)\) with non-trivial, i.e., \(\ne 0,\pm 1\), Brouwer degree.

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References

  1. Bizon, P., Chmaj, T.: Harmonic map between spheres. Proc. R. Soc. Lond. A 453, 403–415 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  2. Eells, J., Sampson, J.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  3. Eells, J., Lemaire, L.: A report on harmonic maps. Bull. Lond. Math. Soc. 10, 1–68 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  4. Eells, J., Lemaire, L.: Another report on harmonic maps. Bull. Lond. Math. Soc. 20, 385–524 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  5. Eells, J., Ratto, A.: Harmonic Maps and Minimal Immersions with Symmetries. Annals of Mathematics Studies, vol. 130. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  6. Foscolo, L., Haskins, M.: New \(\text{G}_2\)-holonomy cones and exotic nearly Kähler structures on \(\mathbb{S}^6\) and \(\mathbb{S}^3\times \mathbb{S}^3\). arXiv:1501.07838

  7. Gastel, A.: Torus equivariant harmonic maps between spheres. Topology 41, 213–227 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gastel, A.: On the harmonic Hopf construction. Proc. Am. Math. Soc. 132, 607–615 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Püttmann, T.: Cohomogeneity one manifolds and self-maps of nontrivial degree. Transform. Groups 14, 225–247 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. Püttmann, T., Siffert, A.: New equivariant harmonic maps between cohomogeneity one manifolds (in preparation)

  11. Siffert, A.: Infinite families of harmonic self-maps of spheres. J. Differ. Equ. 260, 2898–2925 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Smith, R.T.: Harmonic mappings of spheres. Am. J. Math. 97, 364–395 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  13. Urakawa, H.: Equivariant harmonic maps between compact Riemannian manifolds of cohomogeneity one. Mich. Math. J. 40, 27–51 (1993)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgements

It is a pleasure to thank Wolfgang Ziller for making me aware of the Theorem of Malgrange. Furthermore, I would like to thank him for the many conservations during the last year and for the wonderful time I had at the University of Pennsylvania. The author would like to thank Deutsche Forschungsgemeinschaft for supporting this work with the grant SI 2077/1-1. Furthermore, I would also like to thank the Max Planck Institute for Mathematics for the support and the excellent working conditions.

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  1. Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111, Bonn, Germany

    Anna Siffert

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  1. Anna Siffert
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Correspondence to Anna Siffert.

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Siffert, A. Harmonic Self-Maps of \(\mathrm {SU}(3)\) . J Geom Anal 28, 587–605 (2018). https://doi.org/10.1007/s12220-017-9833-0

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  • DOI: https://doi.org/10.1007/s12220-017-9833-0

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