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

#### Abstract

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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