مشخصات پژوهش

Biharmonic hypersurface of Euclidean spaces is a well-known subject in mathematical physics which has been introduced by Bang-Yen Chen and followed by many researchers ([7, 8, 9, 10]). An Euclidean hypersurface x : M^n ----> E^{n+1} is called biharmonic if (D ^2)x = 0, where D is the Laplace operator. The well-known Chen's conjecture states that any biharmonic Euclidean submanifold x : M^n ----> E^{n+1} (satisfying the biharmonicity condition, (D ^2)x = 0) is minimal (i.e. has null mean curvature). Recently, this subject has been extended to a higher order biharmonicity of Euclidean hypersurfaces which play more important roles in geometry and physics. In this paper, we introduce and verify an advanced version of the conjecture, replacing D by its extension, (L_1)-operator. For any spacelike hypersurface M4 of the 5-dimensional Lorentz-Minkowski space E_1^5 , having assumed that it has just three distinct principal curvatures and constant ordinary mean curvature, we prove that it is 1-maximal (i.e. has null 2-th mean curvature).