Biharmonic Functions on the Classical Compact Simple Lie Groups

  • Sigmundur Gudmundsson
  • Stefano Montaldo
  • Andrea Ratto
Article

Abstract

The main aim of this work is to construct several new families of proper biharmonic functions defined on open subsets of the classical compact simple Lie groups \(\mathbf{SU}(n)\), \(\mathbf{SO}(n)\) and \(\mathbf{Sp}(n)\). We work in a geometric setting which connects our study with the theory of submersive harmonic morphisms. We develop a general duality principle and use this to interpret our new examples on the Euclidean sphere \({\mathbb S}^3\) and on the hyperbolic space \({\mathbb H}^3\).

Keywords

Laplace-Beltrami operator Biharmonic functions Polyharmonic functions Lie groups 

Mathematics Subject Classification

58E20 31A30 35R03 

Notes

Acknowledgements

Work partially supported by: PRID 2015 – Università degli Studi di Cagliari.

References

  1. 1.
    Baird, P., Wood, J.C.: Harmonic Morphisms Between Riemannian Manifolds. The London Mathematical Society Monographs, vol. 29. Oxford University Press, Oxford (2003)CrossRefMATHGoogle Scholar
  2. 2.
    Bérard-Bergery, L.: Sur certaines fibrations d’espaces homogènes riemanniens. Compos. Math. 30, 43–61 (1975)MATHGoogle Scholar
  3. 3.
    Besse, A.L.: Einstein Manifolds, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 10, vol. 3. Springer, Berlin (1987)Google Scholar
  4. 4.
    Caddeo, R.: Riemannian manifolds on which the distance function is biharmonic. Rend. Sem. Mat. Univ. Politec. Torino 40, 93–101 (1982)MathSciNetMATHGoogle Scholar
  5. 5.
    Fuglede, B.: Harmonic morphisms between semi-Riemannian manifolds. Ann. Acad. Sci. Fenn. Math. 21, 31–50 (1996)MathSciNetMATHGoogle Scholar
  6. 6.
    Gudmundsson, S.: The Bibliography of Harmonic Morphisms. www.matematik.lu.se/matematiklu/personal/sigma/harmonic/bibliography.html
  7. 7.
    Gudmundsson, S.: Minimal submanifolds of hyperbolic spaces via harmonic morphisms. Geom. Dedicata 62, 269–279 (1996)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Gudmundsson, S., Sakovich, A.: Harmonic morphisms from the classical compact semisimple Lie groups. Ann. Global Anal. Geom. 33, 343–356 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gudmundsson, S., Svensson, M.: Harmonic morphisms from the Grassmannians and their non-compact duals. Ann. Global Anal. Geom. 30, 313–333 (2006)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Gudmundsson, S., Svensson, M., Ville, M.: Complete minimal submanifolds of compact Lie groups. Math. Z. 282, 993–1005 (2016)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Knapp, A.: Lie groups beyond an introduction. In: Progress in Mathematics , vol. 140, Birkhäuser (2002)Google Scholar
  12. 12.
    Loubeau, E., Ou, Y.-L.: Biharmonic maps and morphisms from conformal mappings. Tohoku Math. J. 62, 55–73 (2010)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Ou, Y.-L.: p-Harmonic morphisms, biharmonic morphisms and nonharmonic biharmonic maps. J. Geom. Phys. 56, 358–374 (2006)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of ScienceUniversity of LundLundSweden
  2. 2.Dipartimento di Matematica e InformaticaUniversità degli Studi di CagliariCagliariItaly
  3. 3.Dipartimento di Matematica e InformaticaUniversità degli Studi di CagliariCagliariItaly

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