$L_1$-biharmonic hypersurfaces with three distinct principal curvatures in Euclidean 5-space

Akram Mohammadpouri, Firooz Pashaie


The well-known Chen's conjecture states that any biharmonic submanifold in the Euclidean space, $x: M^n\rightarrow \E^{n+p}$ (satisfying the biharmonicity condition, $\Delta^2 x=0$) is minimal. It was verified and found true
for Euclidean hypersurfaces in several cases, whereas there are many non-existence results in $\cite{Dim92}$. In this paper introduce and verify an advanced version of the conjecture, replacing $\Delta$ by its extension, $L_1$-operator. For any hypersurface $M$ of the $5$-dimensional Euclidean space, having assumed that it has three distinct principal curvatures and constant ordinary mean curvature, we prove that 1-minimal (i.e. has null $2$-th mean curvature).

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


H. Alencar, M. Batista, { em Hypersurfaces with null higher order mean curvature}, Bull Braz Math

Soc., New Series 41(4), (2010), 481-493.


L. J. Al'{i}as, N. G"{u}rb"{u}z,

{em An extension of Takahashi theorem for the linearized operators of

the higher order mean curvatures}, Geom. Dedicata., 121, (2006),



B. Y. Chen, {em Total Mean Curvature and Submanifolds of Finite

Type}, Series in Pure Mathematics, 2. World Scientific Publishing

Co, Singapore, (2014).


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


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


I. Dimitri'{c}, {em Submanifolds of $E^n$ with harmonic mean curvature vector}, Bull. Inst. Math. Acad. Sin., 20 (1992), 53-65.


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


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


S. M. B. Kashani, {em On some $L_1$-finite type (hyper)surfaces in $mathbb{R}^{n+1}$}, Bull. Korean Math. Soc., 46, 1 (2009), 35-43.


A. Mohammadpouri, F. Pashaie, {em $L_r$-biharmonic hypersurfaces in $E^4$}, submitted.


R. C. Reilly, {em Variational properties of functions of the mean curvatures for hypersurfaces in space forms}, J.Differential Geom., 8, 3 (1973), 465-477.


B. Segre, {em 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.


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