Accessible Unlicensed Requires Authentication Published by De Gruyter January 7, 2018

Biharmonic hypersurfaces in 5-dimensional non-flat space forms

Ram Shankar Gupta, Deepika and A. Sharfuddin
From the journal Advances in Geometry

Abstract

We prove that every biharmonic hypersurface having constant higher order mean curvature Hr for r > 2 in a space form M5(c) is of constant mean curvature. In particular, every such biharmonic hypersurface in 𝕊5(1) has constant mean curvature. There exist no such compact proper biharmonic isoparametric hypersurfaces M in 𝕊5(1) with four distinct principal curvatures. Moreover, there exist no proper biharmonic hypersurfaces in hyperbolic space ℍ5 or in E5 having constant higher order mean curvature Hr for r > 2.

MSC 2010: 53D12; 53C40; 53C42

  1. Communicated by: P. Eberlein

Acknowledgements

The work was communicated when the first author was on deputation as Associate Professor, Department of Mathematics, Central University of Jammu, Sainik colony, Jammu-180011, India. The authors are thankful to the referees for their careful readings and helpful suggestions to improve the paper.

References

[1] A. Arvanitoyeorgos, F. Defever, G. Kaimakamis, V. J. Papantoniou, Biharmonic Lorentz hypersurfaces in E14. Pacific J. Math. 229 (2007), 293–305. MR2276512 Zbl 1153.53011Search in Google Scholar

[2] A. Balmuş, S. Montaldo, C. Oniciuc, Classification results for biharmonic submanifolds in spheres. Israel J. Math. 168 (2008), 201–220. MR2448058 Zbl 1172.58004Search in Google Scholar

[3] A. Balmuş, S. Montaldo, C. Oniciuc, Biharmonic hypersurfaces in 4-dimensional space forms. Math. Nachr. 283 (2010), 1696–1705. MR2560665 Zbl 1210.58013Search in Google Scholar

[4] R. Caddeo, S. Montaldo, C. Oniciuc, Biharmonic submanifolds of S3. Internat. J. Math. 12 (2001), 867–876. MR1863283 Zbl 1111.53302Search in Google Scholar

[5] R. Caddeo, S. Montaldo, C. Oniciuc, Biharmonic submanifolds in spheres. Israel J. Math. 130 (2002), 109–123. MR1919374 Zbl 1038.58011Search in Google Scholar

[6] E. Cartan, Sur des familles remarquables d’hypersurfaces isoparamétriques dans les espaces sphériques. Math. Z. 45 (1939), 335–367. MR0000169 Zbl 65.0792.01Search in Google Scholar

[7] B.-Y. Chen, Some open problems and conjectures on submanifolds of finite type. Soochow J. Math. 17 (1991), 169–188. MR1143504 Zbl 0749.53037Search in Google Scholar

[8] B.-Y. Chen, Classification of marginally trapped Lorentzian flat surfaces in E24 and its application to biharmonic surfaces. J. Math. Anal. Appl. 340 (2008), 861–875. MR2390893 Zbl 1160.53007Search in Google Scholar

[9] B.-Y. Chen, Total mean curvature and submanifolds of finite type, volume 27 of Series in Pure Mathematics. World Scientific Publishing Co., Hackensack, NJ 2015. MR3362186 Zbl 1326.53004Search in Google Scholar

[10] B.-Y. Chen, S. Ishikawa, Biharmonic surfaces in pseudo-Euclidean spaces. Mem. Fac. Sci. Kyushu Univ. Ser. A45 (1991), 323–347. MR1133117 Zbl 0757.53009Search in Google Scholar

[11] B.-Y. Chen, S. Ishikawa, Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces. Kyushu J. Math. 52 (1998), 167–185. MR1609044 Zbl 0892.53012Search in Google Scholar

[12] B.-Y. Chen, M. I. Munteanu, Biharmonic ideal hypersurfaces in Euclidean spaces. Differential Geom. Appl. 31 (2013), 1–16. MR3010073 Zbl 1260.53017Search in Google Scholar

[13] F. Defever, G. Kaimakamis, V. Papantoniou, Biharmonic hypersurfaces of the 4-dimensional semi-Euclidean space Es4. J. Math. Anal. Appl. 315 (2006), 276–286. MR2196546 Zbl 1091.53038Search in Google Scholar

[14] I. M. Dimitric, Quadric representation and submanifolds of finite type. PhD thesis, Michigan State University, 1989.Search in Google Scholar

[15] I. Dimitrić, Submanifolds of Em with harmonic mean curvature vector. Bull. Inst. Math. Acad. Sinica20 (1992), 53–65. MR1166218 Zbl 0778.53046Search in Google Scholar

[16] J. Eells, Jr., J. H. Sampson, Harmonic mappings of Riemannian manifolds. Amer. J. Math. 86 (1964), 109–160. MR0164306 Zbl 0122.40102Search in Google Scholar

[17] Y. Fu, Biharmonic hypersurfaces with three distinct principal curvatures in Euclidean 5-space. J. Geom. Phys. 75 (2014), 113–119. MR3126938 Zbl 1283.53005Search in Google Scholar

[18] Y. Fu, Biharmonic hypersurfaces with three distinct principal curvatures in spheres. Math. Nachr. 288 (2015), 763–774. MR3345102 Zbl 1321.53065Search in Google Scholar

[19] R. S. Gupta, On bi-harmonic hypersurfaces in Euclidean space of arbitrary dimension. Glasg. Math. J. 57 (2015), 633–642. MR3395337 Zbl 1323.53065Search in Google Scholar

[20] T. Hasanis, T. Vlachos, Hypersurfaces in E4 with harmonic mean curvature vector field. Math. Nachr. 172 (1995), 145–169. MR1330627 Zbl 0839.53007Search in Google Scholar

[21] T. Ichiyama, J.-i. Inoguchi, H. Urakawa, Classifications and isolation phenomena of bi-harmonic maps and bi-Yang-Mills fields. Note Mat. 30 (2010), 15–48. MR2943022 Zbl 1244.58006Search in Google Scholar

[22] C. Oniciuc, Biharmonic maps between Riemannian manifolds. An. Ştiinţ. Univ. Al. I. Cuza laşi. Mat. (N.S.) 48 (2002), 237–248 (2003). MR2004799 Zbl 1061.58015Search in Google Scholar

[23] P. J. Ryan, Homogeneity and some curvature conditions for hypersurfaces. Tôhoku Math. J. (2) 21 (1969), 363–388. MR0253243 Zbl 0185.49904Search in Google Scholar

Received: 2015-02-20
Revised: 2015-07-25
Revised: 2017-04-28
Published Online: 2018-01-07
Published in Print: 2019-04-24

© 2019 Walter de Gruyter GmbH, Berlin/Boston