## Tohoku Mathematical Journal

### Biharmonic hypersurfaces with three distinct principal curvatures in Euclidean space

Yu Fu

#### Abstract

The well known Chen's conjecture on biharmonic submanifolds states that a biharmonic submanifold in a Euclidean space is a minimal one ([10--13, 16, 18--21, 8]). For the case of hypersurfaces, we know that Chen's conjecture is true for biharmonic surfaces in $\mathbb E^3$ ([10], [24]), biharmonic hypersurfaces in $\mathbb E^4$ ([23]), and biharmonic hypersurfaces in $\mathbb E^m$ with at most two distinct principal curvature ([21]). The most recent work of Chen-Munteanu [18] shows that Chen's conjecture is true for $\delta(2)$-ideal hypersurfaces in $\mathbb E^m$, where a $\delta(2)$-ideal hypersurface is a hypersurface whose principal curvatures take three special values: $\lambda_1, \lambda_2$ and $\lambda_1+\lambda_2$. In this paper, we prove that Chen's conjecture is true for hypersurfaces with three distinct principal curvatures in $\mathbb E^m$ with arbitrary dimension, thus, extend all the above-mentioned results. As an application we also show that Chen's conjecture is true for $O(p)\times O(q)$-invariant hypersurfaces in Euclidean space $\mathbb E^{p+q}$.

#### Article information

Source
Tohoku Math. J. (2), Volume 67, Number 3 (2015), 465-479.

Dates
First available in Project Euclid: 6 November 2015

https://projecteuclid.org/euclid.tmj/1446818561

Digital Object Identifier
doi:10.2748/tmj/1446818561

Mathematical Reviews number (MathSciNet)
MR3420554

Zentralblatt MATH identifier
1283.53005

Subjects
Primary: 53D12: Lagrangian submanifolds; Maslov index

#### Citation

Fu, Yu. Biharmonic hypersurfaces with three distinct principal curvatures in Euclidean space. Tohoku Math. J. (2) 67 (2015), no. 3, 465--479. doi:10.2748/tmj/1446818561. https://projecteuclid.org/euclid.tmj/1446818561

#### References

• K. Akutagawa and S. Maeta, Biharmonic properly immersed submanifolds in Euclidean spaces, Geom. Dedicata 164 (2013), 351–355.
• H. Alencar, A. Barros, O. Palmas, J. G. Reyes and W. Santos, $O(m)\times O(n)$-invariant minimal hypersurfaces in $\mathbb R^{m+n}$, Ann. Global Anal. Geom. 27 (2005), 179–199.
• L. J. Alías, S. C. García-Martínez and M. Rigoli, Biharmonic hypersurfaces in complete Riemannian manifolds, Pacific J. Math. 263 (2013), 1–12.
• A. Arvanitoyeorgos, F. Defever, G. Kaimakamis and V. Papantoniou, Biharmonic Lorentz hypersurfaces in $\mathbb E_1^4$, Pacific J. Math. 229 (2007), no. 2, 293–305.
• A. Balmus, Biharmonic maps and submanifolds, PhD thesis, Universita degli Studi di Cagliari, Italy, 2007.
• A. Balmus, S. Montaldo and C. Oniciuc, Classification results for biharmonic submanifolds in spheres, Israel J. Math. 168 (2008), 201–220.
• A. Balmus, S. Montaldo and C. Oniciuc, Biharmonic hypersurfaces in 4-dimensional space forms, Math. Nachr. 283 (2010), no. 12, 1696–1705.
• R. Caddeo, S. Montaldo and C. Oniciuc, Biharmonic submanifolds of $\mathbb S^3$, Internat. J. Math. 12 (2001), no. 8, 867–876.
• R. Caddeo, S. Montaldo and C. Oniciuc, Biharmonic submanifolds in spheres, Israel J. Math. 130 (2002), 109–123.
• B. Y. Chen, Some open problems and conjectures on submanifolds of finite type, Soochow J. Math. 17 (1991), no. 2, 169–188.
• B. Y. Chen, Pseudo-Riemannian Geometry, $\delta$-invariants and Applications. Word Scientific, Hackensack, NJ, 2011.
• 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, some open problems and conjectures on submanifolds of finite type: recent development, Tamkang J. Math. 45 (1) (2014), 87–108.
• B. Y. Chen, Geometry of Submanifolds, Dekker, New York, 1973.
• B. Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, World Scientific, New Jersey, 1984.
• 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-Euclidean spaces, Kyushu J. Math. 52 (1998), 1–18.
• B. Y. Chen and M. I. Munteanu, Biharmonic ideal hypersurfaces in Euclidean spaces, Differential Geom. Appl. 31 (2013), 1–16.
• F. Defever, Hypersurfaces of $\mathbb E^4$ with harmonic mean curvature vector, Math. Nachr. 196 (1998), 61–69.
• I. Dimitrić, Quadric representation and submanifolds of finite type, Doctoral thesis, Michigan State University, 1989.
• I. Dimitrić, Submanifolds of $\mathbb E^n$ with harmonic mean curvature vector, Bull. Inst. Math. Acad. Sin. 20 (1992), 53–65.
• Y. Fu, Biharmonic hypersurfaces with three dinstinct principal curvatures in $\mathbb E^5$, J. Geom. Phys. 75 (2014), 113–119.
• T. Hasanis and T. Vlachos, Hypersurfaces in $\mathbb E^4$ with harmonic mean curvature vector field, Math. Nachr. 172 (1995), 145–169.
• G. Y. Jiang, 2-Harmonic maps and their first and second variational formulas, Chin. Ann. Math. Ser. A 7 (1986), 389–402.
• G. Y. Jiang, Some non-existence theorems of 2-harmonic isometric immersions into Euclidean spaces, Chin. Ann. Math. Ser. A 8 (1987), 376–383.
• 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, On the generalized Chen's conjecture on biharmonic submanifolds, Michigan Math. J. 61 (2012), 531–542.