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Abstrakty
In decade eighty, Bang-Yen Chen introduced the concept of biharmonic hypersurface in the Euclidean space. An isometrically immersed hypersurface x : Mn → En+1 is said to be biharmonic if ∆2x = 0, where ∆ is the Laplace operator. We study the Lr-biharmonic hypersurfaces as a generalization of biharmonic ones, where Lr is the linearized operator of the (r + 1)th mean curvature of the hypersurface and in special case we have L0 = ∆. We prove that Lr-biharmonic hypersurface of Lr-finite type and also Lr-biharmonic hypersurface with at most two distinct principal curvatures in Euclidean spaces are r-minimal.
Czasopismo
Rocznik
Tom
Strony
91--104
Opis fizyczny
Bibliogr. 20 poz.
Twórcy
autor
- Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
- Department of Mathematics, Faculty of Basic Sciences, University of Maragheh, P.O.Box 55181-83111, Maragheh, Iran
autor
- Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran
- Department of Mathematics, Faculty of Basic Sciences, University of Maragheh, P.O.Box 55181-83111, Maragheh, Iran
Bibliografia
- [1] G.B. Airy, On the strains in the interior of beams, Philos. Trans. R. Soc. London Ser. A 153 (1863) 49-79.
- [2] K. Akutagawa, S. Maeta, Biharmonic properly immersed submanifolds in euclidean spaces, Geom. Dedicata 164 (2013) 351-355.
- [3] H. Alencar, M. Batista, Hypersurfaces with null higher order mean curvature, Bull. Braz. Math. Soc., New Series 41(4) (2010) 481-493.
- [4] L.J. Alías, N. Gürbüz, An Extension of takahashi theorem for the linearized operators of the higher order mean curvatures, Geom. Dedicata 121 (2006) 113-127.
- [5] M. Aminian, S.M.B. Kashani, Lk-biharmonic hypersurfaces in euclidean space, Taiwan J. Math., DOI: 10.11650/tjm.18.2014.4830, (2014) 113-127.
- [6] B.Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, Series in Pure Mathematics, 2. World Scientific Publishing Co, Singapore, 2014.
- [7] I. Dimitrić, Quadric Representation and Submanifolds of Finite Type, Doctoral Thesis, Michigan State University, 1989.
- [8] I. Dimitrić, Submanifolds of En with harmonic mean curvature vector, Bull. Inst. Math. Acad. Sin. 20 (1992) 53-65.
- [9] J. Eells, J.C. Wood, Restrictions on harmonic maps of surfaces, Topology 15 (1976) 263-266.
- [10] T. Hasanis, T. Vlachos, Hypersurfaces in E4 with harmonic mean curvature vector field, Math. Nachr. 172 (1995) 145-169.
- [11] S.M.B. Kashani, On some Li -finite type (hyper)surfaces in Rn+1, Bull. Korean Math. Soc. 46, 1 (2009) 35-43.
- [12] S. Kobayashi, K. Nomizu, Foundations of Differential Geometry, Vol. I, Wiley-Interscience, New York, (1963) Vol. II, (1969).
- [13] J.C. Maxwell, On reciprocal diagrams in space, and their relation to Airy’s function of stress, Proc. London. Math. Soc. 2 (1868) 102.
- [14] S. Nishikawa, Y. Maeda, Conformally flat hypersurfaces in a conformally at Riemannian manifold, Tohoku Math. J. 26 (1974) 159-168.
- [15] T. Otsuki, Minimal Hypersurfaces in a Riemannian Manifold of Constant Curvature, Amer. J. Math. 92, 1 (1970) 145-173.
- [16] H. Reckziegel, On the eigenvalues of the shape operator of an isometric immersion into a space of constant curvature. Math. Ann. 243 (1979) 71-82.
- [17] R.C. Reilly, Variational properties of functions of the mean curvatures for hypersurfaces in space forms, J. Differential Geom. 8, 3 (1973) 465-477.
- [18] B. Segre, Famiglie di ipersuperficie isoparametrische negli spazi euclidei ad un qualunque numero di dimensioni, Atti. Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur 27 (1938) 203-207.
- [19] G. Wei, Complete hypersurfaces in a Eculidean curvature, Diff. Geom. Appl. 26 (2008) 298-306.
- [20] B.G. Yang, X.M. Liu, r-minimal hypersurfaces in space Rn+1 with constant mth mean space forms, Journal of Geometry and Physics 59 (2009) 685-692.
Uwagi
PL
Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).
Typ dokumentu
Bibliografia
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