Taiwanese Journal of Mathematics


M. Aminian and S. M. B. Kashani

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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.

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

First available in Project Euclid: 4 July 2017

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Zentralblatt MATH identifier

Primary: 53C40: Global submanifolds [See also 53B25] 53C42: Immersions (minimal, prescribed curvature, tight, etc.) [See also 49Q05, 49Q10, 53A10, 57R40, 57R42]

$L_k$-operator biharmonic Chen conjecture


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

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  • 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.