×

\(L_k\)-biharmonic hypersurfaces in space forms. (English) Zbl 1373.53072

Summary: In this paper, we introduce \(L_k\)-biharmonic hypersurfaces \(M\) in simply connected space forms \(R^{n+1}(c)\) and propose \(L_k\)-conjecture for them. For \(c=0,-1\), we prove the conjecture when hypersurface \(M\) has two principal curvatures with multiplicities \(1,n-1\), or \(M\) is weakly convex, or \(M\) is complete with some constraints on it and on \(L_k\). We also show that neither there is any \(L_k\)-biharmonic hypersurface \(M^n\) in \( \mathbb {H}^{n+1} \) with two principal curvatures of multiplicities greater than one, nor any \(L_k\)-biharmonic compact hypersurface \(M^n\) in \( \mathbb {R}^{n+1} \) or in \( \mathbb {H}^{n+1} \). As a by-product, we get two useful, important variational formulas. The paper is a sequel to our previous paper, [Taiwanese J. Math. 19, No. 3, 861–874 (2015; Zbl 1357.53056)] in this context.

MSC:

53C40 Global submanifolds
53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53C30 Differential geometry of homogeneous manifolds

Citations:

Zbl 1357.53056
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Akutagawa, K., Maeta, S.: Biharmonic properly immersed submanifolds in Euclidean spaces. Geom. Ded. 164, 351-355 (2013) · Zbl 1268.53068 · doi:10.1007/s10711-012-9778-1
[2] Alías, L.J., García-Martínez, S.C., Rigoli, M.: Biharmonic hypersurfaces in complete Riemannian manifolds. Pacific. J. Math. 263, 1-12 (2013) · Zbl 1278.53059 · doi:10.2140/pjm.2013.263.1
[3] Alías, L. J., Gürbüz, N.: An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures. Geom. Ded. 121, 113-127 (2006) · Zbl 1119.53004 · doi:10.1007/s10711-006-9093-9
[4] Alías, L.J., Kashani, S.M.B.: Hypersurfaces in space forms satisfying the condition Lkx=Ax+b. Taiwan. J. Math. 14, 1957-1977 (2010) · Zbl 1221.53085
[5] Aminian, M., Kashani, S.M.B.: Lk-biharmonic hypersurfaces in the Euclidean space. Taiwan. J. Math. 19, 861-874 (2015) · Zbl 1357.53056 · doi:10.11650/tjm.19.2015.4830
[6] Alías, L.J., Kurose, T., Solanes, G.: Hadamard-type theorems for hypersurfaces in hyperbolic spaces Differ. Geom. Appl. 24, 492-502 (2006) · Zbl 1103.52006 · doi:10.1016/j.difgeo.2006.02.008
[7] Balmuş, A.: Biharmonic Maps and Submanifolds. Geometry Balkan Press. Bucharest, Romania (2009) · Zbl 1201.53072
[8] Balmuş, A., Montaldo, S., Oniciuc, C.: Biharmonic hypersurfaces in 4-dimensional space forms. Math. Nachr. 283, 1696-1705 (2010) · Zbl 1210.58013 · doi:10.1002/mana.200710176
[9] Balmuş, A., Montaldo, S., Oniciuc, C.: Classification results for biharmonic submanifolds in spheres. arXiv:0701155v1[math.DG] (2007) · Zbl 1172.58004
[10] Caddeo, R., Montaldo, S., Oniciuc, C.: Biharmonic submanifolds of S3. Inter. J. Math. 12, 867-876 (2001) · Zbl 1111.53302 · doi:10.1142/S0129167X01001027
[11] Cartan, É.: Familles de surfaces isoparamétriques dans les espaces à courbure constante. Ann. di Mat. 17, 177-191 (1938) · JFM 64.1361.02 · doi:10.1007/BF02410700
[12] Chen, B.Y.: Recent developments of biharmonic conjecture and modified biharmonic conjectures. arXiv:1307.0245v3[math.DG] (2013) · Zbl 1300.53013
[13] Chen, B.Y.: Some open problems and conjectures on submanifolds of finite type. Soochow J. Math. 17, 169-188 (1991) · Zbl 0749.53037
[14] Chen, B.Y.: Some open problems and conjectures on submanifolds of finite type: recent development. Tamkang J. Math. 45, 87-108 (2014) · Zbl 1287.53044 · doi:10.5556/j.tkjm.45.2014.1564
[15] Chen, B.Y.: Total Mean Curvature and Submanifold of Finite Type, 2nd ed. Series in Pure Math, vol. 27. World Scientific, New Jersey (2014) · Zbl 1305.53024
[16] Chen, B.Y., Ishikawa, S.: Biharmonic surfaces in pseudo-Euclidean spaces. Mem. Fac. Sci. Kyushu Univ. 45A, 323-347 (1991) · Zbl 0757.53009
[17] Chen, B.Y., Munteanu, M.I.: Biharmonic ideal hypersurfaces in Euclidean spaces. Differ. Geom. Appl. 31, 1-16 (2013) · Zbl 1260.53017 · doi:10.1016/j.difgeo.2012.10.008
[18] Defever, F.: Hypersurfaces of E \(4 \mathbb{E}^4\) with harmonic mean curvature vector. Math. Nachr. 196, 61-69 (1998) · Zbl 0944.53005 · doi:10.1002/mana.19981960104
[19] Dimitrić, I.: Quadratic representation and submanifolds of finite type, Ph.D. thesis. Michigan State Univ., Lansing MI (1989) · Zbl 0628.58008
[20] Dimitrić, I.: Submanifolds of Em \(\mathbb{E}^m\) with harmonic mean curvature vector. Bull. Inst. Math. Acad. Sinica 20, 53-65 (1992) · Zbl 0778.53046
[21] Eells, J., Sampson, J.H.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109-160 (1964) · Zbl 0122.40102 · doi:10.2307/2373037
[22] Hasanis, T., Vlachos, T.: Hypersurfaces in E \(4 \mathbb{E}^4\) with harmonic mean curvature vector field. Math. Nachr. 172, 145-169 (1995) · Zbl 0839.53007 · doi:10.1002/mana.19951720112
[23] Jiang, G.Y.: 2-harmonic maps and their first and second variational formulas. Chinese Ann. Math. 7A, 388-402 (1986) ; the English translation, Note di Mathematica 28, 209-232 (2008) · Zbl 0628.58008
[24] Luo, Y.: Weakly convex biharmonic hypersurfaces in nonpositive curvature space forms are minimal. Results. Math. 65, 49-56 (2014) · Zbl 1305.53024 · doi:10.1007/s00025-013-0328-4
[25] Lawson, H.: Local rigidity theorems for minimal hypersurfaces. Ann. Math. 89, 187-197 (1969) · Zbl 0174.24901 · doi:10.2307/1970816
[26] Nakauchi, N., Urakawa, H.: Biharmonic submanifolds in a Riemannian manifold with nonpositive curvature. Results. Math. 63, 467-471 (2013) · Zbl 1261.58011 · doi:10.1007/s00025-011-0209-7
[27] O’Neill, B.: Semi-Riemannian Geometry: with Applications to Relativity. Pure Appl. Math. Acad Press, New York (1983) · Zbl 0531.53051
[28] Ou, Y.L.: Biharmonic hypersurfaces in Riemannian manifolds. Pac. J. Math. 248, 217-232 (2010) · Zbl 1205.53066 · doi:10.2140/pjm.2010.248.217
[29] Ou, Y.L., Tang, L.: The generalized Chen’s conjecture on biharmonic submanifolds is false. Mich. Math. J. 61, 531-542 (2012) · Zbl 1268.58015 · doi:10.1307/mmj/1347040257
[30] Reilly, R. C.: Variational properties of functions of the mean curvatures for hypersurfaces in space forms. J. Differ. Geom. 8, 465-477 (1973) · Zbl 0277.53030 · doi:10.4310/jdg/1214431802
[31] Rosenberg, H.: Hypersurfaces of constant curvature in space forms. Bull. Sci. Math. 117, 211-239 (1993) · Zbl 0787.53046
[32] Ryan, P.J.: Homogeneity and some curvature conditions for hypersurfaces. Tohoku Math. J. 21, 363-388 (1969) · Zbl 0185.49904 · doi:10.2748/tmj/1178242949
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.