Taiwanese Journal of Mathematics

$L_k$-BIHARMONIC HYPERSURFACES IN THE EUCLIDEAN SPACE

Abstract

Chen conjecture states that every Euclidean biharmonic submanifold is minimal. In this paper we consider the Chen conjecture for $L_k$-operators. The new conjecture ($L_k$-conjecture) is formulated as follows: If $L_k^{2}x=0$ then $H_{k+1}=0$ where $x:M^{n}\rightarrow\Bbb{R}^{n+1}$ is an isometric immersion of a Riemannian manifold $M^n$ into the Euclidean space $\Bbb{R}^{n+1}$, $H_{k+1}$ is the $(k+1)$-th mean curvature of $M$, and $L_k$ is the linearized operator of the $(k+1)$-th mean curvature of the Euclidean hypersurface $M$. We prove the $L_k$-conjecture for the hypersurface $M$ with at most two principal curvatures.

Article information

Source
Taiwanese J. Math., Volume 19, Number 3 (2015), 861-874.

Dates
First available in Project Euclid: 4 July 2017

https://projecteuclid.org/euclid.twjm/1499133666

Digital Object Identifier
doi:10.11650/tjm.19.2015.4830

Mathematical Reviews number (MathSciNet)
MR3353257

Zentralblatt MATH identifier
1357.53056

Citation

Aminian, M.; Kashani, S. M. B. $L_k$-BIHARMONIC HYPERSURFACES IN THE EUCLIDEAN SPACE. Taiwanese J. Math. 19 (2015), no. 3, 861--874. doi:10.11650/tjm.19.2015.4830. https://projecteuclid.org/euclid.twjm/1499133666

References

• K. Akutagawa and S. Maeta, Biharmonic properly immersed submanifolds in Euclidean spaces, Geom. Dedicata, 164 (2013), 351-355.
• L. J. Al\ptmrs ías, S. C. de Almeida and A. Brasil Jr., Hypersurfaces with constant mean curvature and two principal curvatures in $\mathbb{S}^{n+1}$, An. Acad. Brasil. Cienc., 76 (2004), 489-497.
• L. J. Al\ptmrs ías, S. C. Garc\ptmrs ía-Mart\ptmrs ínez and M. Rigoli, Biharmonic hypersurfaces in complete Riemannian manifolds, Pacific. J. Math., 263(1) (2013), 1-12.
• L. J. Al\ptmrs ías and N. Gürbüz, An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures, Geom. Dedicata, 121 (2006), 113-127.
• L. J. Al\ptmrs ías and S. M. B. Kashani, Hypersurfaces in space forms satisfying the condition $L_kx=Ax+b$, Taiwanese J. Math., 14 (2010), 1957-1978.
• A. Arvanitoyeorgos, F. Defever, G. Kaimakamis and V. J. Papantoniou, Biharmonic Lorentz hypersurfaces in $E_1^4$, Pacific J. Math., 229(2) (2007), 293-305.
• B. Y. Chen, Some open problems and conjectures on submanifolds of finite type, Soochow J. Math., 17(2) (1991), 169-188.
• B. Y. Chen, Some open problems and conjectures on submanifolds of finite type: recent development, Tamkang J. Math., 45(1) (2014), 87-108.
• B. Y. Chen, Total Mean Curvature and Submanifold of Finite Type, 2nd ed., Series in Pure Math., Vol. 27, World Scientific, 2014.
• B. Y. Chen and S. Ishikawa, Biharmonic pseudo-Riemannian submanifolds in pseudo-Euclidean spaces, Kyushu J. Math., 52(1) (1998), 167-185.
• B. Y. Chen and S. Ishikawa, Biharmonic surfaces in pseudo-Euclidean spaces, Mem. Fac. Sci. Kyushu Univ. Ser. A, 45(2) (1991), 323-347.
• B. Y. Chen and M. I. Munteanu, Biharmonic ideal hypersurfaces in Euclidean spaces, Diff. Geom. Appl., 31 (2013), 1-16.
• F. Defever, Hypersurfaces of $E^4$ with harmonic mean curvature vector, Math. Nachr., 196 (1998), 61-69.
• I. Dimitrić, Quadratic Representation and Submanifolds of Finite Type, Ph.D. thesis, Michigan State Univ., Lansing, MI, 1989.
• I. Dimitrić, Submanifolds of $E^m$ with harmonic mean curvature vector, Bull. Inst. Math. Acad. Sinica, 20 (1992), 53-65.
• T. Hasanis and T. Vlachos, Hypersurfaces in $E^4$ with harmonic mean curvature vector field, Math. Nachr., 172 (1995), 145-169.
• S. M. B. Kashani, On some $L_1$-finite type (hyper)surfaces in $\mathbb{R}^{n+1}$, Bull. Korean Math. Soc., 46(1) (2009), 35-43.
• H. Lawson, Local rigidity theorems for minimal hypersurfaces, Ann. Math., 89 (1969), 187-197.
• N. Nakauchi and H. Urakawa, Biharmonic submanifolds in a Riemannian manifold with non-positive curvature, Results. Math., 63 (2013), 467-474.
• B. O'Neill, Semi-Riemannian Geometry: with Applications to Relativity, Pure and Applied Mathematics, Acad. Press, New York, 1983.
• Y. L. Ou and L. Tang, The generalized Chen's conjecture on biharmonic submanifolds is false, Michigan Math. J., 61 (2012), 531-542.
• R. C. Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Differential Geom., 8 (1973), 465-477.
• H. Rosenberg, Hypersurfaces of constant curvature in space forms, Bull. Sci. Math., 117 (1993), 211-239.
• P. J. Ryan, Homogeneity and some curvature conditions for hypersurfaces, Tohoku Math. J., 21 (1969), 363-388.