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Biharmonic submanifolds of \({\mathbb{C}P^n}\)

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We give some general results on proper-biharmonic submanifolds of a complex space form and, in particular, of the complex projective space. These results are mainly concerned with submanifolds with constant mean curvature or parallel mean curvature vector field. We find the relation between the bitension field of the inclusion of a submanifold \({\bar{M}}\) in \({\mathbb{C}P^n}\) and the bitension field of the inclusion of the corresponding Hopf-tube in \({\mathbb{S}^{2n+1}}\). Using this relation we produce new families of proper-biharmonic submanifolds of \({\mathbb{C}P^n}\). We study the geometry of biharmonic curves of \({\mathbb{C}P^n}\) and we characterize the proper-biharmonic curves in terms of their curvatures and complex torsions.

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  1. 1

    Arslan K., Ezentas R., Murathan C., Sasahara T.: Biharmonic anti-invariant submanifolds in Sasakian space forms. Beiträge Algebra Geom. 48, 191–207 (2007)

  2. 2

    Baird P., Fardoun A., Ouakkas S.: Conformal and semi-conformal biharmonic maps. Ann. Global Anal. Geom. 34, 403–414 (2008)

  3. 3

    Balmuş A., Montaldo S., Oniciuc C.: Classification results for biharmonic submanifolds in spheres. Israel J. Math. 168, 201–220 (2008)

  4. 4

    Balmuş, A., Montaldo, S., Oniciuc, C.: Biharmonic hypersurfaces in 4-dimensional space forms. Math. Nachr. (2009, to appear)

  5. 5

    Caddeo R., Montaldo S., Oniciuc C.: Biharmonic submanifolds of \({\mathbb{S}^3}\). Internat. J. Math. 12, 867–876 (2001)

  6. 6

    Caddeo R., Montaldo S., Oniciuc C.: Biharmonic submanifolds in spheres. Israel J. Math. 130, 109–123 (2002)

  7. 7

    Chang S.-Y.A., Wang L., Yang P.: A regularity theory of biharmonic maps. Comm. Pure Appl. Math. 52, 1113–1137 (1999)

  8. 8

    Chen B.Y.: A report on submanifolds of finite type. Soochow J. Math. 22, 117–337 (1996)

  9. 9

    Chiang Y.-J., Sun H.: 2-harmonic totally real submanifolds in a complex projective space. Bull. Inst. Math. Acad. Sinica 27, 99–107 (1999)

  10. 10

    Chiang Y.-J., Wolak R.A.: Transversally biharmonic maps between foliated Riemannian manifolds. Internat. J. Math. 19, 981–996 (2008)

  11. 11

    Dimitric I.: Submanifolds of \({\mathbb{E}^m}\) with harmonic mean curvature vector. Bull. Inst. Math. Acad. Sinica 20, 53–65 (1992)

  12. 12

    Eells J., Lemaire L.: Selected topics in harmonic maps. Conf. Board. Math. Sci. 50, 85 (1983)

  13. 13

    Fetcu D., Oniciuc C.: Explicit formulas for biharmonic submanifolds in Sasakian space forms. Pac. J. Math. 240, 85–107 (2009)

  14. 14

    Ichiyama T., Inoguchi J., Urakawa H.: Bi-harmonic maps and bi-Yang-Mills fields. Note Math. 28(suppl 1), 233–275 (2008)

  15. 15

    Inoguchi J.: Submanifolds with harmonic mean curvature in contact 3-manifolds. Colloq. Math. 100, 163–179 (2004)

  16. 16

    Jiang G.Y.: 2-harmonic isometric immersions between Riemannian manifolds. Chinese Ann. Math. Ser. A 7, 130–144 (1986)

  17. 17

    Lamm T.: Biharmonic map heat flow into manifolds of nonpositive curvature. Calc. Var. Partial Differ. Equ. 22, 421–445 (2005)

  18. 18

    Lawson H.B.: Rigidity theorems in rank-1 symmetric spaces. J. Differ. Geom. 4, 349–357 (1970)

  19. 19

    Maeda S., Adachi T.: Holomorphic helices in a complex space form. Proc. Am. Math. Soc. 125, 1197–1202 (1997)

  20. 20

    Maeda S., Ohnita Y.: Helical geodesic immersions into complex space forms. Geom. Dedic. 30, 93–114 (1989)

  21. 21

    Montaldo S., Oniciuc C.: A short survey on biharmonic maps between Riemannian manifolds. Rev. Un. Mat. Argentina 47(2), 1–22 (2006)

  22. 22

    Moser R.: A second-order variational problem with a lack of coercivity. Proc. Lond. Math. Soc (3) 96, 199–226 (2008)

  23. 23

    Moser R.: A variational problem pertaining to biharmonic maps. Comm. Partial Differ. Equ. 33, 1654–1689 (2008)

  24. 24

    Ou Y.-L.: p-Harmonic morphisms, biharmonic morphisms, and nonharmonic biharmonic maps. J. Geom. Phys. 56, 358–374 (2006)

  25. 25

    Ou, Y.-L.: Biharmonic hypersurfaces in Riemannian manifolds. arXiv:math.DG/09011507v1

  26. 26

    Sasahara T.: Biharmonic Lagrangian surfaces of constant mean curvature in complex space forms. Glasg. Math. J. 49, 497–507 (2007)

  27. 27

    Zhang, W.: New examples of biharmonic submanifolds in \({\mathbb{C}P^{n}}\) and \({\mathbb{S}^{2n+1}}\). arXiv:math.DG/07053961v1

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Correspondence to S. Montaldo.

Additional information

S. Montaldo was supported by PRIN-2007 (Italy): Riemannian metrics and differentiable Manifolds and C. Oniciuc was partially supported by PCE Grant PNII-2228 (502/2009), Romania.

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Fetcu, D., Loubeau, E., Montaldo, S. et al. Biharmonic submanifolds of \({\mathbb{C}P^n}\) . Math. Z. 266, 505–531 (2010). https://doi.org/10.1007/s00209-009-0582-z

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  • Harmonic maps
  • Biharmonic maps
  • Biharmonic submanifolds

Mathematics Subject Classification (2000)

  • 58E20