Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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LIOUVILLE TYPE THEOREMS FOR TRANSVERSALLY HARMONIC AND BIHARMONIC MAPS

  • Jung, Min Joo (Department of Mathematics and Research Institute for Basic Sciences Jeju National University) ;
  • Jung, Seoung Dal (Department of Mathematics and Research Institute for Basic Sciences Jeju National University)
  • Received : 2016.03.30
  • Published : 2017.05.01
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Abstract

In this paper, we study the Liouville type theorems for transversally harmonic and biharmonic maps on foliated Riemannian manifolds.

Keywords

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

References

  1. J. A. Alvarez Lopez, The basic component of the mean curvature of Riemannian foliations, Ann. Global Anal. Geom. 10 (1992), no. 2, 179-194. https://doi.org/10.1007/BF00130919
  2. P. Baird, A. Fardoun, and S. Ouakkas, Liouville-type theorems for biharmonic maps between Riemannian manifolds, Adv. Calc. Var. 3 (2010), no. 1, 49-68. https://doi.org/10.1515/ACV.2010.003
  3. P. Berard, A note on Bochner type theorems for complete manifolds, Manuscripta Math. 69 (1990), no. 3, 261-266. https://doi.org/10.1007/BF02567924
  4. Y.-J. Chiang and R. A. Wolak, Transversally biharmonic maps between foliated Riemannian manifolds, Internat. J. Math. 19 (2008), no. 8, 981-996. https://doi.org/10.1142/S0129167X08004972
  5. S. Dragomir and A. Tommasoli, Harmonic maps of foliated Riemannian manifolds, Geom. Dedicata 162 (2013), 191-229. https://doi.org/10.1007/s10711-012-9723-3
  6. A. El Kacimi Alaoui and E. Gallego Gomez, Applications harmoniques feuilletees, Illinois J. Math. 40 (1996), no. 1, 115-122.
  7. M. J. Jung and S. D. Jung, On transversally harmonic maps of foliated Riemannian manifolds, J. Korean Math. Soc. 49 (2012), no. 5, 977-991. https://doi.org/10.4134/JKMS.2012.49.5.977
  8. S. D. Jung, Harmonic maps of complete Riemannian manifolds, Nihonkai Math. J. 8 (1997), no. 2, 147-154.
  9. S. D. Jung, The first eigenvalue of the transversal Dirac operator, J. Geom. Phys. 39 (2001), no. 3, 253-264. https://doi.org/10.1016/S0393-0440(01)00014-6
  10. S. D. Jung, Variation formulas for transversally harmonic and biharmonic maps, J. Geom. Phys. 70 (2013), 9-20. https://doi.org/10.1016/j.geomphys.2013.03.012
  11. S. D. Jung, K. R. Lee, and K. Richardson, Generalized Obata theorem and its applications on foliations, J. Math. Anal. Appl. 376 (2011), no. 1, 129-135. https://doi.org/10.1016/j.jmaa.2010.10.022
  12. F. W. Kamber and P. Tondeur, Infinitesimal automorphisms and second variation of the energy for harmonic foliations, Tohoku Math. J. 34 (1982), no. 4, 525-538. https://doi.org/10.2748/tmj/1178229154
  13. F. W. Kamber and P. Tondeur, De Rham-Hodge theory for Riemannian foliations, Math. Ann. 277 (1987), no. 3, 415-431. https://doi.org/10.1007/BF01458323
  14. J. Konderak and R. A. Wolak, Transversally harmonic maps between manifolds with Riemannian foliations, Q. J. Math. 54 (2003), no. 3, 335-354. https://doi.org/10.1093/qmath/hag019
  15. J. Konderak and R. A. Wolak, Some remarks on transversally harmonic maps, Glasg. Math. J. 50 (2008), no. 1, 1-16. https://doi.org/10.1017/S001708950700403X
  16. J. S. Pak and S. D. Jung, A transversal Dirac operator and some vanishing theorems on a complete foliated Riemannian manifold, Math. J. Toyama Univ. 16 (1993), 97-108.
  17. E. Park and K. Richardson, The basic Laplacian of a Riemannian foliation, Amer. J. Math. 118 (1996), no. 6, 1249-1275. https://doi.org/10.1353/ajm.1996.0053
  18. R. Schoen and S. T. Yau, Harmonic maps and the topology of stable hypersurfaces and manifolds of nonnegative Ricci curvature, Comm. Math. Helv. 51 (1976), no. 3, 333-341. https://doi.org/10.1007/BF02568161
  19. P. Tondeur, Geometry of foliations, Basel: Birkhauser Verlag, 1997.
  20. S. T. Yau, Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (1975), 201-228. https://doi.org/10.1002/cpa.3160280203
  21. S. T. Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J. 25 (1976), no. 7, 659-670. https://doi.org/10.1512/iumj.1976.25.25051
  22. S. Yorozu, Notes on suqare-integrable cohomology spaces on certain foliated manifolds, Trans. Amer. Math. Soc. 255 (1979), 329-341. https://doi.org/10.1090/S0002-9947-1979-0542884-6

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