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Stability and the index of biharmonic hypersurfaces in a Riemannian manifold

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Abstract

In this paper, we give an explicit second variation formula for a biharmonic hypersurface in a Riemannian manifold similar to that of a minimal hypersurface. We then use the second variation formula to compute the normal stability index of the known biharmonic hypersurfaces in a Euclidean sphere and to prove the nonexistence of unstable proper biharmonic hypersurface in a Euclidean space or a hyperbolic space, which adds another special case to support Chen’s conjecture on biharmonic submanifolds.

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Acknowledgements

The author would like to thank the referee for the comments and suggestions that help to clarify the notion of normal stability and normal index studied in the paper.

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  1. Department of Mathematics, Texas A & M University-Commerce, Commerce, 75429, TX, USA

    Ye-Lin Ou

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  1. Ye-Lin Ou
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This work was supported by a grant from the Simons Foundation (\(\#427231\), Ye-Lin Ou).

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Ou, YL. Stability and the index of biharmonic hypersurfaces in a Riemannian manifold. Annali di Matematica 201, 733–742 (2022). https://doi.org/10.1007/s10231-021-01135-0

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  • DOI: https://doi.org/10.1007/s10231-021-01135-0

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