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Biharmonic submanifolds in manifolds with bounded curvature

Shun Maeta

Shun Maeta

Department of Mathematics, Shimane University, Nishikawatsu 1060, Matsue 690-8504, Japan

E-mail Address: shun.maeta@gmail.com

E-mail Address: maeta@riko.shimane-u.ac.j

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https://doi.org/10.1142/S0129167X16500890Cited by:1 (Source: Crossref)

Abstract

We consider a complete biharmonic submanifold $ϕ : (M, g) \to (N, h)$ $ϕ : (M, g) \to (N, h)$ in a Riemannian manifold with sectional curvature bounded from above by a non-negative constant $c$ $c$ . Assume that the mean curvature is bounded from below by $\sqrt{c}$ $\sqrt{c}$ . If (i) $\int_{M} {(| H |^{2} - c)}^{p} d v_{g} < \infty$ $\int_{M} {(| H |^{2} - c)}^{p} d v_{g} < \infty$ , for some $0 < p < \infty$ $0 < p < \infty$ , or (ii) the Ricci curvature of $M$ $M$ is bounded from below, then the mean curvature is $\sqrt{c}$ $\sqrt{c}$ . Furthermore, if $M$ $M$ is compact, then we obtain the same result without the assumption (i) or (ii). These are affirmative partial answers to Balmuş–Montaldo–Oniciuc conjecture.

Keywords:

AMSC: 53C43, 58E20, 53C40

References

1. K. Akutagawa and S. Maeta, Biharmonic properly immersed submanifolds in Euclidean spaces, Geom. Dedicata 164 (2013) 351–355. Crossref, Web of Science, Google Scholar
2. A. Balmuş, S. Montaldo and C. Oniciuc, New results toward the classification of biharmonic submanifolds in $S^{n}$ $S^{n}$ , An. Stiint. Univ. “Ovidius” Constanta Ser. Mat. 20(2) (2012) 89–114. Google Scholar
3. A. Balmuş, S. Montaldo and C. Oniciuc, Biharmonic PNMC submanifolds in spheres, Ark. Mat. 51(2) (2013) 197–221. Crossref, Web of Science, Google Scholar
4. A. Balmuş, S. Montaldo and C. Oniciuc, Properties of biharmonic submanifolds in spheres, J. Geom. Symmetry Phys. 17 (2010) 87–102. Google Scholar
5. A. Balmuş, S. Montaldo and C. Oniciuc, Biharmonic hypersurfaces in 4-dimensional space forms, Math. Nachr. 283 (2010) 1696–1705. Crossref, Web of Science, Google Scholar
6. A. Balmuş, S. Montaldo and C. Oniciuc, Classification results and new examples of proper-biharmonic submanifolds in spheres, Note Mat. 28 (2008) 49–61. Google Scholar
7. A. Balmuş, S. Montaldo and C. Oniciuc, Classification results for biharmonic submanifolds in spheres, Israel J. Math. 168 (2008) 201–220. Crossref, Web of Science, Google Scholar
8. A. Balmuş and C. Oniciuc, Biharmonic submanifolds with parallel mean curvature vector field in spheres, J. Math. Anal. Appl. 386 (2012) 619–630. Crossref, Web of Science, Google Scholar
9. A. Balmuş and C. Oniciuc, Biharmonic surfaces of $S^{4}$ $S^{4}$ , Kyushu J. Math. 63 (2009) 339–345. Crossref, Web of Science, Google Scholar
10. R. Caddeo, S. Montaldo and C. Oniciuc, Biharmonic submanifolds in spheres, Israel J. Math. 130 (2002) 109–123. Crossref, Web of Science, Google Scholar
11. R. Caddeo, S. Montaldo and C. Oniciuc, Biharmonic submanifolds of $S^{3}$ $S^{3}$ , Internat. J. Math. 12 (2001) 867–876. Link, Web of Science, Google Scholar
12. R. Caddeo, S. Montaldo, C. Oniciuc and P. Piu, Surfaces in three-dimensional space forms with divergence-free stress-bienergy tensor, Ann. Mat. 193 (2014) 529–550. Crossref, Web of Science, Google Scholar
13. S.-Y. Cheng and S.-T. Yau, Maximal space-like hypersurfaces in the Lorentz–Minkowski spaces, Ann. of Math. 104 (1976) 407–419. Crossref, Web of Science, Google Scholar
14. J. H. Chen, Compact 2-harmonic hypersurfaces in $S^{n + 1}$ $S^{n + 1}$ (1), Acta Math. Sinica 36 (1993) 49–56. Google Scholar
15. J. Eells and L. Lemaire, Selected topics in harmonic maps, CBMS, Amer. Math. Soc. 50 (1983). Google Scholar
16. J. Eells and J. H. Sampson, Harmonic mappings of Riemannian manifolds, Amer. J Math. 86 (1964) 109–160. Crossref, Web of Science, Google Scholar
17. D. Fetcu, E. Loubeau, S. Montaldo and C. Oniciuc, Biharmonic submanifolds of CPⁿ, Math. Z. 266 (2010) 505–531. Crossref, Web of Science, Google Scholar
18. T. Ichiyama, J. Inoguchi and H. Urakawa, Classifications and isolation phenomena of bi-harmonic maps and bi-Yang–Mills fields, Note Mat. 30(2) (2010) 15–48. Google Scholar
19. T. Ichiyama, J. Inoguchi and H. Urakawa, Bi-harmonic maps and bi-Yang–Mills fields, Note Mat. 28 (2008) 233–275. Google Scholar
20. J. Inoguchi, Biminimal submanifolds in contact 3-manifolds, Balkan. J. Geom. 12(1) (2007) 56–67. Web of Science, Google Scholar
21. G. Y. Jiang, $2$ $2$ -harmonic maps and their first and second variational formulas, Chinese Ann. Math. 7A (1986) 388–402; the English translation, Note Mat. 28 (2008) 209–232. Google Scholar
22. N. Koiso and H. Urakawa, Biharmonic submanifolds in a Riemannian manifold, arXiv:1408.5494v1[math.DG]. Google Scholar
23. E. Loubeau, S. Montaldo and C. Oniciuc, The stress–energy tensor for biharmonic maps, Math. Z. 259 (2008) 503–524. Crossref, Web of Science, Google Scholar
24. E. Loubeau and S. Montaldo, Biminimal immersion, Proc. Edinb. Math. Soc. 51 (2008) 421–437. Crossref, Web of Science, Google Scholar
25. E. Loubeau and C. Oniciuc, On the biharmonic and harmonic indices of the Hopf map, Trans. Amer. Math. Soc. 359(11) (2007) 5239–5256. Crossref, Web of Science, Google Scholar
26. Y. Luo, On biharmonic submanifolds in non-positively curved manifolds, J. Geom. Phys. 88 (2015) 76–87. Crossref, Web of Science, Google Scholar
27. Y. Luo, Weakly convex biharmonic hypersurfaces in nonpositive curvature space forms are minimal, Results Math. 65 (2014) 49–56. Crossref, Web of Science, Google Scholar
28. S. Maeta, Biharmonic maps from a complete Riemannian manifold into a non-positively curved manifold, Ann. Global Anal. Geom. 46 (2014) 75–85. Crossref, Web of Science, Google Scholar
29. S. Maeta, Properly immersed submanifolds in complete Riemannian manifolds, Adv. Math. 253 (2014) 139–151. Crossref, Web of Science, Google Scholar
30. S. Maeta and H. Urakawa, Biharmonic Lagrangian submanifolds in Kähler manifolds, Glasg. Math. J. 55 (2013) 465–480. Crossref, Web of Science, Google Scholar
31. N. Nakauchi and H. Urakawa, Biharmonic submanifolds in a Riemannian manifold with non-positive curvature, Results Math. 63 (2013) 467–474. Crossref, Web of Science, Google Scholar
32. Y.-L. Ou and L. Tang, On the generalized Chen’s conjecture on biharmonic submanifolds, Michigan Math. J. 61 (2012) 531–542. Crossref, Web of Science, Google Scholar
33. X. F. Wang and L. Wu, Proper biharmonic submanifolds in a sphere, Acta Math. Sin. (Engl. Ser.) 28 (2012) 205–218. Crossref, Web of Science, Google Scholar
34. Z.-P. Wang and Y.-L. Ou, Biharmonic Riemannian submersions from 3-manifolds, Math. Z. 269 (2011) 917–925. Crossref, Web of Science, Google Scholar
35. G. Wheeler, Chen’s conjecture and $ϵ$ $ϵ$ -superbiharmonic submanifolds of Riemannian manifolds, Internat. J. Math. 24 (2013) 6, Article ID: 1350028. Link, Web of Science, Google Scholar