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L k -biharmonic Hypersurfaces in Space Forms

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Abstract

In this paper, we introduce L k -biharmonic hypersurfaces M in simply connected space forms R n+1(c) and propose L k -conjecture for them. For c=0,−1, we prove the conjecture when hypersurface M has two principal curvatures with multiplicities 1,n−1, or M is weakly convex, or M is complete with some constraints on it and on L k . We also show that neither there is any L k -biharmonic hypersurface M n in \( \mathbb {H}^{n+1} \) with two principal curvatures of multiplicities greater than one, nor any L k -biharmonic compact hypersurface M n in \( \mathbb {R}^{n+1} \) or in \( \mathbb {H}^{n+1} \). As a by-product, we get two useful, important variational formulas. The paper is a sequel to our previous paper, (Taiwan. J. Math., 19, 861–874, 5) in this context.

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Authors and Affiliations

  1. Department of Pure Mathematics, Tarbiat Modares University, Tehran, Iran

    M. Aminian

  2. Department of Pure Mathematics Faculty of Mathematics Sciences, Tarbiat Modares University, P. O. Box 14115-134, Tehran, Iran

    S. M. B. Kashani

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  1. M. Aminian
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  2. S. M. B. Kashani
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Correspondence to S. M. B. Kashani.

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Aminian, M., B. Kashani, S.M. L k -biharmonic Hypersurfaces in Space Forms. Acta Math Vietnam 42, 471–490 (2017). https://doi.org/10.1007/s40306-016-0195-7

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  • DOI: https://doi.org/10.1007/s40306-016-0195-7

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