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arXiv · 2006.07612

On Chen's biharmonic conjecture for hypersurfaces in $\mathbb R^5$

Yu Fu·Min-Chun Hong·Xin Zhan
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Abstract

A longstanding conjecture on biharmonic submanifolds, proposed by Chen in 1991, is that {\it any biharmonic submanifold in a Euclidean space is minimal}. In the case of a hypersurface $M^n$ in $\mathbb R^{n+1}$, Chen's conjecture was settled in the case of $n=2$ by Chen and Jiang around 1987 independently. Hasanis and Vlachos in 1995 settled Chen's conjecture for a hypersurface with $n=3$. However, the general Chen's conjecture on a hypersurface $M^n$ remains open for $n> 3$. In this paper, we settle Chen's conjecture for hypersurfaces in $\mathbb R^{5}$ for $n=4$.

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Yu Fu, Min-Chun Hong, Xin Zhan. 2020-06-13. On Chen's biharmonic conjecture for hypersurfaces in $\mathbb R^5$. https://arxiv.org/abs/2006.07612

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