Acta Mathematica Vietnamica

, Volume 42, Issue 3, pp 471–490 | Cite as

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.

Keywords

Lk operator (bi)energy functionals (bi)harmonic maps Chen conjecture 

Mathematics Subject Classification (2010)

53C40 53C42 

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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2016

Authors and Affiliations

  1. 1.Department of Pure MathematicsTarbiat Modares UniversityTehranIran
  2. 2.Department of Pure Mathematics Faculty of Mathematics SciencesTarbiat Modares UniversityTehranIran

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