Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Generalized Chen's Conjecture for Biharmonic Maps on Foliations

  • Xueshan Fu (Department of Mathematics, Shenyang University of Technology) ;
  • Seoung Dal Jung (Department of Mathematics, Jeju National University)
  • Received : 2023.08.31
  • Accepted : 2023.11.03
  • Published : 2023.12.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we prove the generalized Chen's conjecture for (𝓕, 𝓕')-biharmonic maps, such maps are critical points of the transversal bienergy functional.

Keywords

Acknowledgement

The second author was supported by the 2022 scientific promotion program funded by Jeju National University.

References

  1. J. A. Alvarez Lopez, The basic component of the mean curvature of Riemannian foliations, Ann. Global Anal. Geom., 10(1992), 179-194. https://doi.org/10.1007/BF00130919
  2. R. Caddeo, S. Montaldo and P. Piu, On biharmonic maps, Contemp. Math., 288(2001), 286-290. https://doi.org/10.1090/conm/288/04836
  3. B.-Y. Chen, Some open problems and conjectures on submanifolds of finite type, Soochow J. Math., 17(1991), 169-188.
  4. Y. J. Chiang and R. Wolak, Transversally biharmonic maps between foliated Riemannian manifolds, Internat. J. Math., 19(2008), 981-996. https://doi.org/10.1142/S0129167X08004972
  5. D. Dominguez, A tenseness theorem for Riemannian foliations, C. R. Acad. Sci. Paris Ser. I Math., 320(1995), 1331-1335.
  6. 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
  7. X. Fu, J. Qian and S. D. Jung, Harmonic maps on weighted Riemannian foliations, arXiv:2212.05639v2 [math.DG].
  8. M. P. Gaffney, A special Stokes' theorem for complete Riemannian manifold, Ann. of Math., 60(1954), 140-145. https://doi.org/10.2307/1969703
  9. G. Y. Jiang, 2-harmonic maps and their first and second variational formula, Chinese Ann. Math. Ser. A, 7(1986), 389-402.
  10. S. D. Jung, The first eigenvalue of the transversal Dirac operator, J. Geom. Phys., 39(2001), 253-264. https://doi.org/10.1016/S0393-0440(01)00014-6
  11. 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
  12. M. J. Jung and S. D. Jung, On transversally harmonic maps of foliated Riemannian manifolds, J. Korean Math. Soc., 49(2012), 977-991. https://doi.org/10.4134/JKMS.2012.49.5.977
  13. M. J. Jung and S. D. Jung, Liouville type theorems for transversally harmonic and biharmonic maps, J. Korean Math. Soc., 54(2017), 763-772. https://doi.org/10.4134/JKMS.j160208
  14. F. W. Kamber and P. Tondeur, Infinitesimal automorphisms and second variation of the energy for harmonic foliations, Tohoku Math. J., 34(1982), 525-538.
  15. J. J. Konderak and R. A. Wolak, Transversally harmonic maps between manifolds with Riemannian foliations, Q. J. Math., 54(2003), 335-354 https://doi.org/10.1093/qjmath/54.3.335
  16. P. March, M. Min-Oo and E. A. Ruh, Mean curvature of Riemannian foliations, Canad. Math. Bull., 39(1996), 95-105. https://doi.org/10.4153/CMB-1996-012-4
  17. A. Mason, An application of stochastic ows to Riemannian foliations, Houston J. Math., 26(2000), 481-515.
  18. P. Molino, Riemannian foliations, Birkhuser Boston, Inc., Boston, MA, 1988, xii+339 pp.
  19. N. Nakauchi, H. Urakawa and S. Gudmundsson, Biharmonic maps into a Riemannian manifold of non-positive curvature, Geom. Dedicata, 169(2014), 263-272. https://doi.org/10.1007/s10711-013-9854-1
  20. 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.
  21. H. K. Pak and J. H. Park, Transversal harmonic transformations for Riemannian foliations, Ann. Global Anal. Geom., 30(2006), 97-105. https://doi.org/10.1007/s10455-006-9032-x
  22. E. Park and K. Richardson, The basic Laplacian of a Riemannian foliation, Amer. J. Math., 118(1996), 1249-1275. https://doi.org/10.1353/ajm.1996.0053
  23. P. Tondeur, Foliations on Riemannian manifolds, Springer-Verlag, New York, 1988, xii+247 pp.
  24. P. Tondeur, Geometry of foliations, Monogr. Math. 90, Birkhuser Verlag, Basel, 1997, viii+305 pp.
  25. S. T, Yau, Some function-theoretic properties of complete Riemannian manifold and their applications to geometry, Indiana Univ. Math. J., 25(1976), 659-670. https://doi.org/10.1512/iumj.1976.25.25051
  26. S. Yorozu and T. Tanemura, Green's theorem on a foliated Riemannian manifold and its applications, Acta Math. Hungar., 56(1990), 239-245. https://doi.org/10.1007/BF01903838