Biharmonic hypersurfaces of the 4-dimensional semi-Euclidean space \(\mathbb E_s^4\). (English) Zbl 1091.53038
Let \(M\) be a non-degenerate hypersurface of a 4-dimensional semi-Euclidean space \(E\). Suppose that \(M\) is a biharmonic submanifold and its shape operator is diagonalizable. The authors prove that \(M\) must be a minimal submanifold of \(E\). The result is related to the following conjecture of B. Y. Chen: The only biharmonic submanifolds of Euclidean spaces are the minimal ones. Thus, the authors give a partial answer for the extension of Chen’s conjecture to 4-dimensional semi-Euclidean spaces.
Reviewer: Aurel Bejancu (Safat)
MSC:
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |
| 53C43 | Differential geometric aspects of harmonic maps |
References:
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