Skip to main content

New results toward the classification of biharmonic submanifolds in 𝕊n Cover

.blurhash-client-img { display: none !important; }

New results toward the classification of biharmonic submanifolds in 𝕊n

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

Volume 20 (2012): Issue 2 (June 2012)

By: Cristina Flaut

Open Access

|May 2013

Abstract

Abstract We prove some new rigidity results for proper biharmonic immer- sions in Sⁿ of the following types: Dupin hypersurfaces; hypersurfaces, both compact and non-compact, with bounded norm of the second fun- damental form; hypersurfaces satisfying intrinsic properties; PMC sub- manifolds; parallel submanifolds.

References

[1] A. Balmuş, S. Montaldo, C. Oniciuc. Biharmonic PNMC submanifolds in spheres, to appear in Ark. Mat.
Search in Google Scholar
[2] A. Balmuş, S. Montaldo, C. Oniciuc. Properties of biharmonic submani- folds in spheres. J. Geom. Symmetry Phys. 17 (2010), 87 102.
Search in Google Scholar
[3] A. Balmuş, S. Montaldo, C. Oniciuc. Biharmonic hypersurfaces in 4- dimensional space forms. Math. Nachr. 283 (2010), 1696-1705.
Search in Google Scholar
[4] A. Balmuş, S. Montaldo, C. Oniciuc. Classification results and new exam- ples of proper biharmonic submanifolds in spheres. Note Mat. 28 (2008), suppl. 1, 49-61.
Search in Google Scholar
[5] A. Balmuş, S. Montaldo, C. Oniciuc. Classification results for biharmonic submanifolds in spheres. Israel J. Math. 168 (2008), 201-220.
Search in Google Scholar
[6] A. Balmuş, C. Oniciuc. Biharmonic submanifolds with parallel mean cur- vature vector field in spheres. J. Math. Anal. Appi. 386 (2012), 619-630.
Search in Google Scholar
[7] A. Balmuş, C. Oniciuc. Biharmonic surfaces of §⁴. Kyushu .J. Math. 63 (2009), 339-345.10.2206/kyushujm.63.339
Search in Google Scholar
[8] A. Besse. Einstein manifolds. Springer-Verlag, Berlin, 1987.10.1007/978-3-540-74311-8
Search in Google Scholar
[9] R. Caddeo, S. Montaldo, C. Oniciuc. Biharmonic submanifolds in spheres. Israel J. Math. 130 (2002), 109-123.10.1007/BF02764073
Search in Google Scholar
[10] R. Caddeo, S. Montaldo, C. Oniciuc. Biharmonic submanifolds of §³. Internat. J. Math. 12 (2001), 807-876.
Search in Google Scholar
[11] S. Chang. A closed hypersurface with constant scalar and mean curvature in §⁴ is isoparametric. Comm. Anal. Geom 1 (1993), 71-100.10.4310/CAG.1993.v1.n1.a4
Search in Google Scholar
[12] B-Y. Chen. Pseudo-Riemannian geometry, invariants and applications, World Scientific, Hackensack, NJ, 2011.10.1142/8003
Search in Google Scholar
[13] B-Y. Chen. Complete| classification of parallel spatial surfaces in pseudo- Riemannian space forms with arbitrary index and dimension. J. Geom. Phys. 00 (2010), 200-280.10.1016/j.geomphys.2009.09.012
Search in Google Scholar
[14] B-Y. Chen. A report on submanifolds of finite type. Soochow J. Math. 22 (1990), 117-337.
Search in Google Scholar
[15] B-Y. Chen. Some open problems and conjectures on submanifolds of finite type. Soochow J. Math. 17 (1991), 109-188.
Search in Google Scholar
[16] B-Y. Chen. Total Mean Curvature and Submanifolds of Finite Type. Se- ries in Pure Mathematics, 1. World Scientific Publishing Co., Singapore, 1984.10.1142/0065
Search in Google Scholar
[17] J.H. Chen. Compact 2-harmonic hypersurfaces in §ⁿ⁺¹(l). Acta Math. Sinica 30 (1993), 49-50.
Search in Google Scholar
[18] Q.-M. Cheng. Compact hypersurfaces in a unit sphere with infinite fun- damental group. Pacific .1. Math. 212 (2003), 341-347.10.2140/pjm.2003.212.49
Search in Google Scholar
[19] S.S. Chern, M. doCarmo, S. Kobayashi. Minimal submanifolds of a sphere with second fundamental form of constant length. 1970 Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, 111., 1908), 59-75 Springer, New York.10.1007/978-3-642-48272-4_2
Search in Google Scholar
[20] F. Defever. Hypersurfaces of E⁴ with harmonic mean curvature vector. Math. Nachr. 190 (1998), 01-09.
Search in Google Scholar
[21] I. Dimitric. Submanifolds of E^m with harmonic mean curvature vector. Bull. Inst. Math. Acad. Sinica 20 (1992), 53-05.
Search in Google Scholar
[22] J. Erbacher. Reduction of the codimension of an isometric immersion. J. Differ. Geom. 5 (1971), 333-340.10.4310/jdg/1214429997
Search in Google Scholar
[23] D. Fetcu, C. Oniciuc. Biharmonic integral C-parallel submanifolds in 7- dimensional Sasakian space forms, Tohoku Math. J., to appear.
Search in Google Scholar
[24] T. Hasanis, T. Vlachos. Hypersurfaces in E⁴ with harmonic mean curva- ture vector field. Math. Nachr. 172 (1995), 145-169.
Search in Google Scholar
[25] T. Ichiyama, .1.1. Inoguchi, H. Urakawa. Bi-harmonic maps and bi-Yang- Mills fields. Note Mat. 28 suppl. n. 1 (2008), 233-275.
Search in Google Scholar
[26] G.Y. Jiang. 2-harmonic isometric immersions between Riemannian man- ifolds. Chinese Ann. Math. Ser. A 7 (1986), 130-144.
Search in Google Scholar
[27] H.B. Lawson. Local rigidity theorems for minimal hypersurfaces. Ann. of Math. (2) 89 (1969), 187-197.10.2307/1970816
Search in Google Scholar
[28] P. Li. Harmonic functions and applications to complete manifolds. XIV Escola de Geometria Diferencial (IMPA), Rio de Janeiro, 2006.
Search in Google Scholar
[29] N. Nakauchi, H. Urakawa. Biharmonic hypersurfaces in a Riemannian manifold with non-positive Ricci curvature. Ann. Glob. Anal. Geom. 40 (2011), 125-131.
Search in Google Scholar
[30] N. Nakauchi, H. Urakawa. Biharmonic submanifolds in a Riemannian manifold with non-positive curvature. Results Math., to appear.
Search in Google Scholar
[31] C. Oniciuc. Tangency and Harmonicity Properties. PhD The- sis, Geometry Balkan Press 2003, http://www.mathem.pub.ro/-dgds/mono/dgdsmono.htm
Search in Google Scholar
[32] C. Oniciuc. Biharmonic maps between Riemannian manifolds. An. Stiint. Univ. ALL Cuza Iasi Mat (N.S.) 48 (2002), 237-248.
Search in Google Scholar
[33] T[ Sasahara. Legendre surfaces in Sasakian space forms whose mean curvature vectors are eigenvectors. Pubi. Math. Debrecen 67 (2005), 285-303.10.1515/dema-2005-0121
Search in Google Scholar
[34] Y.-L. Ou, L. Tang. On the generalized Chen’s conjecture on biharmonic submanifolds. Michigan Math. J., to appear.
Search in Google Scholar
[35] C. Peng, C. Terng. Minimal hypersurfaces of spheres with constant scalar curvature. Seminar on minimal submanifolds, 177-198, Ann. of Math. Stud., 103, Princeton Univ. Press, Princeton, NJ, 1983.10.1515/9781400881437-010
Search in Google Scholar
[36] W. Santos. Submanifolds with parallel mean curvature vector in spheres. Tohoku Math. J. 46 (1994), 403-415.10.2748/tmj/1178225720
Search in Google Scholar
[37] J. Simons. Minimal varieties in Riemannian manifolds. Ann. of Math. 88 (1968), 62-105.[38] X.F. Wang, L. Wu. Proper biharmonic submanifolds in a sphere. Acta Math. Sin. (Engl. Ser.) 28 (2012), 205-218.10.1007/s10114-012-9018-5
Search in Google Scholar
[39] S.-T. Yau. Harmonic functions on complete Riemannian manifolds. Com- mun. Pure Appi. Math. 28 (1975), 201-228.10.1002/cpa.3160280203
Search in Google Scholar
[40] S.-T. Yau. Submanifolds with constant mean curvature I. Amer. J. Math. 90 (1974), 34G-36G.
Search in Google Scholar
[41] W. Zhang. New examples of biharmonic submanifolds in CPn and S²ⁿ⁺¹. An. Stiint. Univ. Al.I. Cuza Iasi Mat (N.S.) 57 (2011), 207-218. 10.2478/v10157-010-0046-0
Search in Google Scholar

Articles in this issue

DOI: https://doi.org/10.2478/v10309-012-0043-2 | Journal eISSN: 1844-0835 | Journal ISSN: 1224-1784

Journal RSS Feed

Language: English

Page range: 89 - 114

Published on: May 17, 2013

Published by: Ovidius University of Constanta

In partnership with: Paradigm Publishing Services

Publication frequency: 3 issues per year

Keywords:

biharmonic submanifolds,

CMC submanifolds,

parallel submanifolds

Related subjects:

General mathematics

© 2013 , published by Ovidius University of Constanta
This work is licensed under the Creative Commons License.

Volume 20 (2012): Issue 2 (June 2012)

Previous article