Some open problems and conjectures on submanifolds of finite type: recent development

Main Article Content

Bang-Yen Chen

Abstract

Submanifolds of finite type were introduced by the author during the late 1970s. The first results on this subject were collected in author's books [26,29]. In 1991, a list of twelve open problems and three conjectures on finite type submanifolds was published in [40]. A detailed survey of the results, up to 1996, on this subject was given by the author in [48]. Recently, the study of finite type submanifolds, in particular, of biharmonic submanifolds, have received a growing attention with many progresses since the beginning of this century. In this article, we provide a detailed account of recent development on the problems and conjectures listed in [40].

Article Details

How to Cite
Chen, B.-Y. (2014). Some open problems and conjectures on submanifolds of finite type: recent development. Tamkang Journal of Mathematics, 45(1), 87–108. https://doi.org/10.5556/j.tkjm.45.2014.1564
Section
Papers
Author Biography

Bang-Yen Chen, Department of Mathematic, Michigan State University

Department of Mathematics,MichiganStateUniversity

References

K. Akutagawa and S. Maeta, Biharmonic properly immersed submanifolds in Euclidean spaces, Geom. Dedicata, 164 (2013), 351--355.

L. J. Alias, S. C. Garcia-Martinez and M. Rigoli, Biharmonic hypersurfaces in complex Riemannian manifolds, Pacific J. Math., 263(2013), 1--12.

J. Arroyo, O. J. Garay and J. J. Mencia, On a family of surfaces of revolution of finite Chen-type, Kodai Math. J., 21 (1998), 73--80.

K. Arslan, R. Ezentas, C. Murathan and T. Sasahara, Biharmonic anti-invariant submanifolds in Sasakian space forms, Beitrage Algebra Geom., 48 (2007), 191--207.

A. Arvanitoyeorgos, G. Kaimakamis and D. Palamourdas, Chen's conjecture and generalizations, Symp. Diff. Geom. Submanifolds, 109--112, 2007.

C. Baikoussis, D. E. Blair, B. Y. Chen and F. Defever, Hypersurfaces of restricted type in Minkowski space, Geom. Dedicata, 62 (1996), 318--332.

C. Baikoussis, B. Y. Chen and L. Verstraelen, Ruled surfaces and tubes with finite type Gauss maps, Tokyo J. Math., 16(1993), 341--349.

A. Balmucs, S. Montaldo and C. Oniciuc, Classification results for biharmonic submanifolds in spheres, Israel J. Math., 168 (2008), 201--220.

A. Balmucs, S. Montaldo and C. Oniciuc, Classification results and new examples of proper biharmonic

submanifolds in spheres, Note Mat., 1 (2008), suppl. no. 1, 49--61.

A. Balmucs, S. Montaldo and C. Oniciuc, Properties of biharmonic ubmanifolds in spheres, J. Geom. Symmetry Phys., 17(2010), 87--102.

A. Balmucs, S. Montaldo and C. Oniciuc, Biharmonic hypersurfaces in $4$-dimensional space forms, Math. Nachr., 283 (2010), 1696--1705.

A. Balmucs, S. Montaldo and C. Oniciuc, New results toward the classification of biharmonic submanifolds in $S^{n}$, An. St. Univ. Ovidius Constanta, 20 (2012), 89--114.

M. Barros, There exist no 2-type surfaces in $E^3$ which are images under stereographic projection of minimal surfaces in $S^3$, Ann. Global Anal. Geom., 10 (1992), 219--226.

M. Barros and B. Y. Chen, Stationary $2$-type surfaces in a hypersphere, J. Math. Soc. Japan, 39 (1987), 627--648.

M. Barros and B. Y. Chen, Spherical submanifolds which are of 2-type via the second standard immersion of the sphere, Nagoya Math. J., 108(1987), 77--91.

M. Barros and O. J. Garay, $2$-type surfaces in $S^3$, Geometriae Dedicata, 24(1987), 329--336.

M. Barros and O. J. Garay, On submanifolds with harmonic mean curvature, Proc. Amer. Math. Soc.,123 (1996), 545--2549.

M. Barros and F. Urbano, Spectral geometry of minimal surfaces in the sphere, Tohoku Math. J., 39(1987), 575--588.

D. Brada and L. Niglio, Connected compact minimal Chen type--$1$ submanifolds of Grassmannian manifold, Bull. Soc. Math. Belg. Ser. B., 44(1992), 299--310.

R. Caddeo, S. Montaldo and and C. Oniciuc, Biharmonic submanifolds of$S^3$, Internat. J. Math., 12(2001), 867--876.

R. Caddeo, S. Montaldo and and C. Oniciuc, Biharmonic submanifolds in spheres, Israel J. Math., 130(2002), 109--123.

R. Caddeo, S. Montaldo and P. Piu, On biharmonic maps, Contemp. Math., 288 (2001), 286--290.

B. Y. Chen, On the total curvature of immersed manifolds, IV: Spectrum and total mean curvature, Bull. Inst. Math. Acad. Sinica, 7 (1979), 301--311.

B. Y. Chen, On the total curvature of immersed manifolds, VI: Submanifolds of finite type and their applications, Bull. Inst. Math. Acad. Sinica, 11(1983),309--328.

B. Y. Chen, On submanifolds of finite type, Soochow J. Math., (1983), 65--81.

B. Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, World Scientific Publisher, 1984.

B. Y. Chen, On the first eigenvalue of Laplacian of compact minimal submanifolds of rank one symmetric spaces, Chinese J. Math., 11(1983), 259--273.

B. Y. Chen, Finite type submanifolds in pseudo-Euclidean spaces and applications, Kodai Math. J., 8(1985), 358--374.

B. Y. Chen, Finite Type Submanifolds and Generalizations, University of Rome, Rome, 1985.

B. Y. Chen, $2$-type submanifolds and their applications, Chinese J. Math., 14 (1986),1--14.

B. Y. Chen, Finite type pseudo-Riemannian submanifolds, Tamkang J. Math., 17 (1986), 137--151.

B. Y. Chen, Some estimates of total tension and their applications, Kodai Math. J., 10 (1987), 93--101.

B. Y. Chen, Surfaces of finite type in Euclidean $3$-space, Bull. Soc. Math. Belg. Ser. B, 39(1987), 243--254.

B. Y. Chen, Mean curvature of $2$-type spherical submanifolds, Chinese J. Math., 16(1988), 1--9.

B. Y. Chen, Null $2$-type surfaces in $E^3$ are circular cylinders, Kodai Math. J., 11(1988), 295--299.

B. Y. Chen, Null $2$-type surfaces in Euclidean space, Algebra, Analysis and Geometry, (1988), 1--18.

B. Y. Chen, $3$-type surfaces in $S^3$, Bull. Soc. Math. Belg. Ser. B, 42 (1990), 379--381.

Chen, B. Y., Local rigidity theorems of $2$-type hypersurfaces in a hypersphere, Nagoya Math. J., 122 (1991), 139--148.

Chen, B. Y., Linearly independent, orthogonal, and equivariant immersions, Kodai Math. J., 14 (1991), 341--349.

B. Y. Chen, Some open problems and conjectures on submanifolds of finite type, Soochow J. Math., 17(1991), 169--188.

B. Y. Chen, Submanifolds of finite type in hyperbolic spaces, Chinese J. Math., 20 (1992), 5--21.

B. Y. Chen, Some pinching and classification theorems for minimal submanifolds, Arch. Math., 60(1993), 568--578.

B. Y. Chen, Classification of tensor product immersions which are of $1$--type, Glasgow Math. J., 36(1994), 255--264.

B. Y. Chen, Some classification theorems for submanifolds in Minkowski space-time, Archiv der Math., 62(1994), 177--182.

B. Y. Chen, Tubular hypersurfaces satisfying a basic equality, Soochow J. Math., 20(1994), 569--586.

B. Y. Chen, A Riemannian invariant for submanifolds in space forms and its applications, Geometry and topology of submanifolds, VI, 58--81, 1994.

B. Y. Chen, Submanifolds in de Sitter space-time satisfying $Delta H=lambda H$, Iseral J. Math., 89(1995), 373--391.

B. Y. Chen, A report on submanifolds of finite type, Soochow J. Math., 22(1996), 117--337.

B. Y. Chen, Some new obstruction to minimal and Lagrangian isometric immersions, Japan. J. Math., 26(2000), 105--127.

B. Y. Chen, Riemannian submanifolds, Handbook of Differential Geometry, vol. 1, North Holland, Amsterdam, 187--418, 2000.

B. Y. Chen, Submanifolds with parallel mean curvature vector in Riemannian and indefinite space forms, Arab J. Math. Sci., 16 (2010), 1--46.

B. Y. Chen, Pseudo-Riemannian Geometry, {$delta$}-invariants and Applications, World Scientific, Hackensack, NJ, 2011.

B. Y. Chen, A tour through $delta$-invariants: From Nash embedding theorem to ideal immersions, best ways of living and beyond, Publ. Inst. Math. (Beograd) (N.S.), 94(108)(2013), 67--80.

B. Y. Chen, Recent developments of biharmonic conjectures and modified biharmonic conjectures, Pure and Applied Differential Geometry -- PADGE 2012, 81--90, Shaker Verlag, Aachen, 2013.

B. Y. Chen, M. Barros and O. J. Garay, Spherical finite type hypersurfaces, Algebras Groups Geom., 4 (1987), 58--72.

B. Y. Chen, J. Deprez, F. Dillen, L. Verstraelen and L. Vrancken, Finite type curves, Geometry and Topology of Submanifolds, II, 76--110, 1990.

B. Y. Chen, J. Deprez and Verheyen, P., Immersions, dans un espace euclidien, d'un espace symetrique compact de rang una geodesiques simples, C. R. Acad. Sc. Paris, 304(1987), 567--570.

B. Y. Chen, J. Deprez and P. Verheyen, Immersions with many circular geodesics, Geometry and Topology of Submanifolds IV, 111--132, 1992.

B. Y. Chen, J. Deprez and P. Verheyen, Immersions with geodesics of 2--type, Geometry and Topology of Submanifolds, IV, 87--110, 1992.

B. Y. Chen, J. Deprez and P. Verheyen, A note on the centroid set of compact symmetric spaces, Geometry and Topology of Submanifolds, IV, 3--10, 1992.

B. Y. Chen and F. Dillen, Surfaces of finite type and constant curvature in the $3$-sphere, C. R. Math. Rep. Acad. Sci. Canada, 12 (1990), 47--49.

B. Y. Chen and F. Dillen, Quadrics of finite type, J. Geometry, 38 (1990), 16--22.

B. Y. Chen, F. Dillen and H. Song, Quadric hypersurfaces of finite type, Colloq. Math., 63(1992), 145--152.

B. Y. Chen, F. Dillen and L. Verstraelen, Finite type space curves, Soochow J. Math., 12(1986), 1--10.

B. Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, Ruled surfaces of finite type, Bull. Austral. Math. Soc., 42(1990), 447--453.

B. Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, variational minimal principle characterizes submanifolds of finite type, C.R. Acad. Sc. Paris, 317 (1993), 961--965.

B. Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, Submanifolds of restricted type, J. of Geometry, 46(1993), 20--32 .

B. Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, A variational minimal principle and its applications, Kyungpook Math. J., 35(1995), no. 3, Speical Issue,

--444.

B. Y. Chen, F. Dillen, L. Verstraelen and L. Vrancken, Compact hypersurfaces determined by a spectral variational principle, Kyushu J. Math., 49 (1995), 103--121.

B. Y. Chen and O. J. Garay, $delta(2)$-ideal null 2-type hypersurfaces of Euclidean space are spherical cylinders, Kodai Math. J., 35 (2012), 382--391.

B. Y. Chen and S. Ishikawa, Biharmonic surfaces in pseudo-Euclidean spaces, Mem. Fac. Sci. Kyushu Univ. A, 45(1991), 323--347.

B. Y. Chen and S. Ishikawa, Biharmonic pseudo-Riemannian submanifolds in pseudo-Eucli-dean spaces, Kyushu J. Math., 52(1998), 167--185.

B. Y. Chen and S. Ishikawa, On classification of some surfaces of revolution of finite type, Tsukuba J. Math., 17(1993), 287--298.

B. Y. Chen and W. E. Kuan, The cubic representation of a submanifold, Beitrage zur Algebra und Geometrie, 35(1994), 55-66.

B. Y. Chen and S. J. Li, $3$-type hypersurfaces in a hypersphere, Bull. Soc. Math. Belg. Ser. B, 43 (1991), 135--141.

B. Y. Chen and H. S. Lue, Some $2$--type submanifolds and applications, Ann. Fac. Sc. Toulouse Math. Ser. V, 9 (1988), 121--131.

B. Y. Chen, J.-M. Morvan and T. Nore, Energie, tension et order des applications a valeurs dans un espace euclidien, C. R. Acad. Sc. Paris, 301(1985), 123--126.

B. Y. Chen, J.-M. Morvan and T. Nore, Energy, tension and finite type maps, Kodai Math. J., 9 (1986), 406--418.

B. Y. Chen and M.I. Munteanu, Biharmonic ideal hypersurfaces in Euclidean spaces, Differential Geom. Appl., 31(2013), 1--16.

B. Y. Chen and T. Nagano, Harmonic metrics, harmonic tensors, and Gauss maps, J. Math. Soc. Japan, 36(1984), 295--313.

B. Y. Chen and K. Ogiue, On totally real submanifolds, Trans. Amer. Math. Soc., 193(1974), 257--266.

B. Y. Chen and M. Petrovic, On spectral decomposition of immersions of finite type, Bull. Austral. Math. Soc., 44(1991), 117--129.

B. Y. Chen and P. Piccinni, Submanifolds with finite type Gauss map, Bull. Austral. Math. Soc., 44 (1987), 161--186.

B. Y. Chen and H. Song, Null $2$--type surfaces in Minkowski space--time, Algebras Groups Geom., 6(1989), 333--352.

B. Y. Chen and H. Song, Null 2-type surfaces in Minkowski space-time, II, Atti Acad. Peloritana Pericolanti Cl. Sci. Fis. Mat. Natur., 68(1989), 1--12.

B. Y. Chen and L. Verstraelen, Laplace Transformations of Submanifolds, Centre for Pure and Applied Differential Geometry (PADGE), vol. 1. Brussels-Leuven, 1995.

F. Defever, Hypersurfaces of ${mathbb{E}}^4$ with harmonic mean curvature vector, Math. Nachr., 196(1998), 61--69.

F. Defever, R. Deszez and L. Verstraelen, L., The compact cylcides of Dupin and a conjecture of B.-Y. Chen, J. Geometry, 46(1993), 33--38.

F. Defever, R. Deszez and L. Verstraelen, The Chen--type of noncompact cylcides of Dupin, Glasgow Math. J., 36(1994), 71--75.

J. Deprez, Immersions of finite type of compact homogeneous Riemannian manifolds, Doctoral Thesis, Katholieke Universiteit Leuven, 1988.

J. Deprez, Immersions of finite type, Geometry and Topology of Submanifolds II, 111--133, 1990.

J. Deprez, F. Dillen and L. Vrancken, Finite type curves on quadrics, Chinese J. Math., 18 (1990), 95--121.

F. Dillen, Ruled submanifolds of finite type, Proc. Amer. Math. Soc., 114(1992), 795--798.

F. Dillen, J. Pas, and L. Verstraelen, On surfaces of finite type in Euclidean $3$-space, Kodai Math. J. 13(1990), 10--21.

I.Dimitric, Quadric representation and submanifolds of finite type, Ph.D. Thesis, Michigan State University, 1989.

I.Dimitric, Submanifolds of $E^m$ with harmonic mean curvature vector, Bull. Inst. Math. Acad. Sinica, 20 (1992), 53--65.

I.Dimitric, Quadric representation of a submanifold and spectral geometry, Proc. Symp. Pure Math. 54, Part 3 (1993),155--168.

I.Dimitric, Quadric representation of a submanifold, Proc. Amer. Math. Soc., 114 (1992), 201--210 .

I.Dimitric, $1$-type submanifolds of non-Euclidean complex space forms, Bull. Belg. Math. Soc. Simon Stevin, 4(1997), 673--684.

I.Dimitric, $CR$-submanifolds of $HP^m$ and hypersurfaces of the Cayley plane whose Chen-type is $1$, Kyungpook Math. J., 40 (2000), 407--429.

I.Dimitric, Low-type submanifolds of real space forms via the immersions by projector, Differential Geom. Appl., 27(2009), 507--526.

I.Dimitric, Hopf hypersurfaces of low type in non-flat complex space forms, Kodai Math. J., 34 (2011), 202--243.

U. Dursun, Null $2$-type submanifolds of the Euclidean space $mathbb E^5$ with parallel normalized mean curvature vector, Kodai Math. J., 28(2005), 191--198.

U. Dursun, Null $2$-type submanifolds of the Euclidean space $mathbb E^5$ with non-parallel mean curvature vector, J. Geom., 86(2006), 73--80.

U. Dursun, On null $2$-type submanifolds of Euclidean spaces, Int. Electron. J. Geom., 2(2009), no. 2, 20--26.

J. Eells and L. Lemaire, A report on harmonic maps, Bull. London Math. Soc.,10 (1978), 1--68.

J. Eells and L. Lemaire, Another report on harmonic maps, Bull. London Math. Soc. 20 (1988), 385--524.

A. Ferrandez, O. J. Garay and P. Lucas, Finite type ruled manifolds shaped on spherrical submanifolds, Arch. fur Math., 57 (1991), 97--104.

A. Ferrandez, O. J. Garay and P. Lucas, On a certain class of conformally flat Euclidean hypersurfaces, Global differential geometry and global analysis (Berlin, 1990), 48--54, Lecture Notes in Math., 1481, Springer, Berlin, 1991.

A. Ferrandez and P. Lucas, P., Null finite type hypersurfaces in space forms, Kodai Math. J., 14 (1991), no. 3, 406--419.

FLMO D. Fetcu, E. Loubeau, S. Montaldo and C. Oniciuc, Biharmonic submanifolds of $CP^n$, Math. Z., 266(2010), 505--531.

D. Fetcu, and C. Oniciuc, Explicit formulas for biharmonic submanifolds in Sasakian space forms, Pacific J. Math., 240(2009), 85-107.

Y, Fu, Biharmonic and quasi-biharmonic slant surfaces in Lorentzian complex space forms, Abstr. Appl. Anal. 2013, Art. ID 412709, 7 pp.

Y, Fu, Biharmonic submanifolds with parallel mean curvature vector in

pseudo-euclidean spaces, Math. Phys. Anal. Geom., 16 (2013), 331--344.

Y, Fu, Biharmonic hypersurfaces with three distinct principal curvatures in Euclidean $5$-space, J. Geom. Phys., 75 (2014), 113--119.

O. J. Garay, Spherical Chen surfaces which are mass-symmetric and of $2$-type, J. Geometry, 33 (1988), 39--52.

O. J. Garay, Finite type cones shaped on spherical submanifolds, Proc. Amer. Math. Soc., 104 (1988), 868--870.

O. J. Garay, An extension of Takahashi's theorem, Geom. Dedicata, 34 (1990), 105--112.

T. Hasanis and T. Vlachos, A local classification of 2-type surfaces in $S^3$, Proc. Amer. Math. Soc., 122 (1991), 533--538.

T. Hasanis and T. Vlachos, Spherical $2$-type hypersurfaces, J. Geometry, 40 (1991), 82--94.

T. Hasanis and T. Vlachos, Hypersurfaces of $E^{n+1}$ satisfying $Delta x=Ax+B$, J. Austral. Math. Soc. A., 53 (1992), 377--384.

T. Hasanis and T. Vlachos, Surfaces of finite type with constant mean curvature, Kodai Math. J., 16(1993), 244--352.

T. Hasanis and T. Vlachos, Hypersurfaces in ${mathbb{E}}^4$ with harmonic mean curvature vector field, Math. Nachr., 172(1995), 145--169.

T. Hasanis and T. Vlachos, Hypersurfaces with constant scalar curvature and constant mean curvature, Ann. Global Anal. Geom., 13 (1995), 69--77.

C. S. Houh, Rotation surfaces of finite type, Algebras Groups Geom., 7(1990), 199--209.

C. S. Houh and S. J. Li, Generalized Chen submanifolds, J. Geometry, 48 (1993), 144--156.

S. Ishikawa, Classification problems of finite type submanifolds and biharmonic submanifolds, Doctoral Thesis, Kyushu University, 1992.

C. Jang, Some linearly independent immersions into their adjoint hyperquadrics, J. Korean Math. Soc., 33 (1996), 169--181.

G. Y. Jiang, $2$-harmonic isometric immersions between Riemannian manifolds, Chinese Ann. Math. Ser. A, 7(1986), 130--144.

S. J. Li, Null $2$-type surfaces in Em with parallel normalized mean curvature vector, Math. J. Toyama Univ., 17(1994), 23--30.

S. J. Li, Null $2$-type Chen surfaces, Glasgow Math. J., 37(1995), 233--242.

T. Liang and Y.-L. Ou, Biharmonic hypersurfaces in a conformally flat space, Results Math., 64 (2013), 91--104.

Y. Luo, Weakly convex biharmonic hypersurfaces in Euclidean spaces are minimal, arXiv: 1305.71981v1, 2013.

Y. Luo, On biharmonic submanifolds in non-positively curved manifolds and $epsilon$-superbi-harmonic submanifolds, airXiv:1306.6069v1, 2013.

Y. Luo, On the Willmore energy and Chen's conjecture, preprint, 2013.

M. Kotani, A decomposition theorem of 2-type immersions}, Nagoya Math. J., 118(1990), 55--66.

S. Maeta, $k$-harmonic maps into a Riemannian manifold with constant sectional curvature, Proc. Amer. Math. Soc., 140 (2012), 1635--1847.

S. Maeta, Biminimal properly immersed submanifolds in the Euclidean spaces, J. Geom. Phys., 62(2012), 2288--2293.

S. Maeta, Biminimal properly immersed submanifolds in complete Riemannian manifold of non-positive curvature, airXiv:1208.0473v2, 2012.

S. Maeta, Biharmonic maps from a complete Riemannian manifold into a non-positively curved manifold, ArXiv:1305.7065v1, 2013.

S. Maeta and H. Urakawa, Biharmonic Lagrangian submanifolds in Kahler manifolds, Glasgow Math. J., 55(2013), 465--480.

A. Martinez and A. Ros, On real hypersurfaces of finite type of $CP^m$, Kodai Math. J., 7 (1984), 304--316.

Y. Miyata, $2$-type surfaces of constant curvature in $S^n$, Tokyo J. Math., {bf 11} (1988), 157--204.

S. Montaldo and A. Ratto, Biharmonic submanifolds into quadrics, arXiv:1309.0631v1, 2013.

S. Montaldo and A. Ratto, Biharmonic submanifolds into ellipsoids, arXiv:1309.1601v1, 2013.

Y. Nagatomo, Finite type hypersurfaces of a sphere, Tokyo J. Math., 14(1991), 85--92.

N. Nakauchi and H. Urakawa, Biharmonic hypersurfaces in a Riemannian manifold with non-positive Ricci curvature, Ann. Glob. Anal. Geom., 40 (2011), 125--131.

N. Nakauchi and H. Urakawa, Biharmonic submanifolds in a Riemannian manifold with non-positive curvature, Results Math. 63(2013), 467--471.

N. Nakauchi, H. Urakawa and S. Gudmundsson, Biharmonic maps into a Riemannian manifold of non-positive curvature, Geom. Dedicata, DOI 10.1007/s10711-013-9854-1.

Y.-L. Ou, On conformal biharmonic immersions, Ann. Global Anal. Geom., 36 (2009), 133--142.

Y.-L. Ou, Biharmonic hypersurfaces in Riemannian manifolds, Pacific J. Math., 248 (2010) 217--232.

Y.-L. Ou, Some constructions of biharmonic maps and Chen's conjecture on biharmonic hypersurfaces, J. Geom. Phys., 62 (2012), 751--762.

Y.-L. Ou and L. Tang, The generalized Chen's conjecture on biharmonic submanifolds is false, Michigan Math. J., 61 (2012), 531--542.

B. Rouxel, Chen submanifolds, Geometry and Topology of Submanifolds, VI, 185--198, 1994.

A. Ros, Spectral geometry of CR-minimal submanifolds in the complex projective space, Kodai Math. J., 6 (1983), 88--99.

A. Ros, On spectral geometry of Kaehler submanifolds, J. Math. Soc. Japan, 36 (1984), 433--448.

A. Ros, Curvature pinching and eigenvalue rigidity for minimal submanifolds, Math. Z., 191(1986), 537--548.

T. Sasahara, Stability of biharmonic Legendrian submanifolds in Sasakian space forms, Canad. Math. Bull., 51(2008), 448--459.

T. Sasahara, A class of biminimal Legendrian submanifolds in Sasaki space forms, Math. Nachr., Doi 10.1002/mana.201200153, 2013.

T. Tahakashi, Minimal immersions of Riemannian manifolds, J. Math. Soc. Japan, 18 (1966), 380--385.

S. Udagawa, Bi-order real hypersurfaces in complex projective space, Kodai Math. J., 10(1987), 182-196.

H. Urakawa, Sasaki manifolds, Kahler cone manifolds and biharmonic submanifolds, arXiv:1306.6123v2, 2013.

L. Verstraelen, On submanifolds of finite Chen type and of restricted type, Results in Math. 20 (1991), 744-755.

L. Verstraelen, Curves and surfaces of finite Chen type, Geometry and Topology of Submanifolds III, 304--314, 1991.

X. F. Wang and L. Wu, Proper biharmonic submanifolds in a sphere, Acta Math. Sin. (Engl. Ser.), 28 (2012), 205--218.

Z.-P. Wang, Y.-L. Ou and H.-C. Yang, Biharmonic maps from $2$-sphere, arXiv: 1310.0562v2, 2013.

G. Wheeler, Chen's conjecture and $epsilon$-superbiharmonic submanifolds of Riemannian manifolds, Intern. J. Math., 24 (2013), no. 4, 1350028 (6 pages).