Document Type : Research Paper

**Authors**

Dept. of Math, Rafsanjan University of Vali-e-Asr, Iran

10.22103/jmmrc.2021.15736.1116

**Abstract**

In this paper we classify proper $L_k$-biharmonic hypersurfaces $ M $, in the unit Euclidean sphere which has two principal curvatures and we show that they are open pieces of standard products of spheres. Also we study proper $L_k$-biharmonic compact hypersurfaces $ M $ with respect to $tr(S^2\circ P_k)$ and $ H_k $ where $ S $ is the shape operator, $ P_k $ is the Newton transformation and $ H_k $ is the $ k $-th mean curvature of $ M $, and by definiteness's assumption of $ P_k $, we show that $ H_{k+1} $ is constant.

**Keywords**

Ded., 164:351–355, 2013.

[2] L. J. Al´ıas, S. C. Garc´ıa-Mart´ınez, and M. Rigoli. Biharmonic hypersurfaces in complete Riemannian

manifolds. Pacific. J. Math, 263:1–12, 2013.

[3] L. J. Al´ıas and N. G¨urb¨uz. An extension of Takahashi theorem for the linearized operators of the

higher order mean curvatures. Geom. Ded., 121:113–127, 2006.

[4] L. J. Al´ıas and S. M. B. Kashani. Hypersurfaces in space forms satisfying the condition lkx = ax+b.

Taiwanese J.M., 14:1957–1977, 2010.

[5] M. Aminian. Introduction of T-harmonic maps. to appear in Pure Appl. Math.

[6] M. Aminian. Lk-biharmonic hypersurfaces in space forms with three distinct principal curvatures.

Commun. Korean Math. Soc., 35(4):1221–1244, 2020.

[7] M. Aminian and S. M. B. Kashani. Lk-biharmonic hypersurfaces in the Euclidean space. Taiwanese

J.M., 19:861–874, 2015.

[8] M. Aminian and S. M. B. Kashani. Lk-biharmonic hypersurfaces in space forms. Acta Math. Vietnam.,

42:471–490, 2017.

[9] M. Aminian and M. Namjoo. fLk-harmonic maps and fLk-harmonic morphisms. Acta Math. Vietnam.,

2020.

[10] A. Balmus¸, S. Montaldo, and C. Oniciuc. Classification results and new examples of proper biharmonic

submanifolds in spheres. Note Mat., suppl. n. 1:49–61, 2008.

[11] A. Balmus¸, S. Montaldo, and C. Oniciuc. New results toward the classification of biharmonic submanifolds

in Sn. An. S¸t. Univ. Ovidius Constant¸a, 20:89–114, 2012.

[12] J. L. M. Barbosa and A. G. Colares. Stability of hypersurfaces with constant r-mean curvature. Ann.

GlobalAnal. Geom., 15:277–297, 1997.

[13] R. Caddeo, S. Montaldo, and C. Oniciuc. Biharmonic submanifolds of S3. Inter. J. Math., 12:867–876,

2001.

[14] B. Y. Chen. Some open problems and conjectures on submanifolds of finite type. Soochow J. Math.,

17:169–188, 1991.

[15] B. Y. Chen. Total Mean Curvature and Submanifold of Finite Type. Series in Pure Math. World Scientific,

2nd edition, 2014.

[16] B. Y. Chen and M. I. Munteanu. Biharmonic ideal hypersurfaces in Euclidean spaces. Diff. Geom.

Appl., 31:1–16, 2013.

[17] S. Y. Cheng and S. T. Yau. Hypersurfaces with constant scalar curvature. Math. Ann., 225:195–204,

1977.

[18] F. Defever. Hypersurfaces of E4 with harmonic mean curvature vector. Math. Nachr., 196:61–69,

1998.

[19] I. Dimitri´c. Submanifolds of Em with harmonic mean curvature vector. Bull. Inst. Math. Acad. Sinica,

20:53–65, 1992.

[20] J. Eells and J. H. Sampson. Harmonic mappings of Riemannian manifolds. Amer. J. Math., 86:109–

160, 1964.

[21] T. Hasanis and T. Vlachos. Hypersurfaces in E4 with harmonic mean curvature vector field. Math.

Nachr., 172:145–169, 1995.

[22] G. Y. Jiang. 2-harmonic maps and their first and second variational formulas. Chinese Ann. Math.,

7A:388–402, 1986. the English translation, Note di Mathematica, 28 (2008) 209-232.

[23] N. Nakauchi and H. Urakawa. Biharmonic submanifolds in a Riemannian manifold with nonpositive

curvature. Results. Math., 63:467–471, 2013.

[24] Y. L. Ou. Biharmonic hypersurfaces in Riemannian manifolds. Pacific J. Math., 248:217–232, 2010.

[25] Y. L. Ou and L. Tang. On the generalized chen’s conjecture on biharmonic submanifolds. Michigan

Math. J., 61:531–542, 2012.

[26] R. C. Reilly. Variational properties of functions of the mean curvatures for hypersurfaces in space

forms. J. Diff. Geom., 8:465–477, 1973.

[27] H. Rosenberg. Hypersurfaces of constant curvature in space forms. Bull. Sci. Math., 117:211–239,

1993.

Winter and Spring 2021

Pages 69-78

**Receive Date:**14 April 2020**Revise Date:**07 March 2021**Accept Date:**15 April 2021