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Advances in Geometry

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Biharmonic hypersurfaces in 5-dimensional non-flat space forms

Ram Shankar Gupta
  • Corresponding author
  • University School of Basic and Applied Sciences, Guru Gobind Singh Indraprastha University, Sector-16C, Dwarka, New Delhi-110078, India
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/ Deepika
  • Department of Mathematics, Faculty of Natural Sciences, Jamia Millia Islamia (Central University), New Delhi-110025, India
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/ A. Sharfuddin
  • Department of Mathematics, Faculty of Natural Sciences, Jamia Millia Islamia (Central University), New Delhi-110025, India
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Published Online: 2018-01-07 | DOI: https://doi.org/10.1515/advgeom-2017-0019

Abstract

We prove that every biharmonic hypersurface having constant higher order mean curvature Hr for r > 2 in a space form M5(c) is of constant mean curvature. In particular, every such biharmonic hypersurface in ūĚēä5(1) has constant mean curvature. There exist no such compact proper biharmonic isoparametric hypersurfaces M in ūĚēä5(1) with four distinct principal curvatures. Moreover, there exist no proper biharmonic hypersurfaces in hyperbolic space ‚Ąć5 or in E5 having constant higher order mean curvature Hr for r > 2.

Keywords: Biharmonic submanifolds; mean curvature vector; isoparametric hypersurfaces

MSC 2010: 53D12; 53C40; 53C42

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About the article


Received: 2015-02-20

Revised: 2015-07-25

Revised: 2017-04-28

Published Online: 2018-01-07


Citation Information: Advances in Geometry, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2017-0019.

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