The well known Chen's conjecture on biharmonic submanifolds states that abiharmonic 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 istrue for biharmonic surfaces in $\mathbb E^3$ ([10], [24]), biharmonichypersurfaces in $\mathbb E^4$ ([23]), and biharmonic hypersurfaces in $\mathbbE^m$ with at most two distinct principal curvatures ([21]). The most recentwork of Chen-Munteanu [18] shows that Chen's conjecture is true for$\delta(2)$-ideal hypersurfaces in $\mathbb E^m$, where a $\delta(2)$-idealhypersurface is a hypersurface whose principal curvatures take three specialvalues: $\lambda_1, \lambda_2$ and $\lambda_1+\lambda_2$. In this paper, weprove that Chen's conjecture is true for hypersurfaces with three distinctprincipal curvatures in $\mathbb E^m$ with arbitrary dimension, thus, extendall the above-mentioned results. As an application we also show that Chen'sconjecture is true for $O(p)\times O(q)$-invariant hypersurfaces in Euclideanspace $\mathbb E^{p+q}$.
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机译：著名的Chen关于双调和子流形的猜想指出，欧几里得空间中的双调和子流形是最小的（[10-13，16,18-21，8]）。对于超曲面，我们知道Chen猜想对于$ \ mathbb E ^ 3 $（[10]，[24]）中的双谐波曲面，$ \ mathbb E ^ 4 $（[23]）中的双谐波超曲面是正确的。 $ \ mathbbE ^ m $中的超曲面具有最多两个不同的主曲率（[21]）。 Chen-Munteanu [18]的最新工作表明，对于$ \ mathbb E ^ m $中的$ \ delta（2）$理想超曲面，Chen的猜想是正确的，其中$ \ delta（2）$理想超曲面是超曲面其主曲率采用三个特殊值：$ \ lambda_1，\ lambda_2 $和$ \ lambda_1 + \ lambda_2 $。在本文中，我们证明了Chen的猜想对于具有任意维度的\ mathbb E ^ m $中具有三个不同主曲率的超曲面是正确的，因此扩展了上述所有结果。作为一个应用，我们还证明了Chen的猜想对于欧氏空间$ \ mathbb E ^ {p + q} $中的$ O（p）\乘以O（q）$不变超曲面是正确的。
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