• Title/Summary/Keyword: linear Weingarten hypersurfaces

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A SHARP INTEGRAL INEQUALITY FOR COMPACT LINEAR WEINGARTEN HYPERSURFACES

  • de Lima, Henrique F.;dos Santos, Fabio R.;Rocha, Lucas S.
    • Bulletin of the Korean Mathematical Society
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    • v.59 no.3
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    • pp.789-799
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    • 2022
  • We establish a sharp integral inequality related to compact (without boundary) linear Weingarten hypersurfaces (immersed) in a locally symmetric Einstein manifold and we apply it to characterize totally umbilical hypersurfaces and isoparametric hypersurfaces with two distinct principal curvatures, one which is simple, in such an ambient space. Our approach is based on the ideas and techniques introduced by Alías and Meléndez in [4] for the case of hypersurfaces with constant scalar curvature in an Euclidean round sphere.

LINEAR WEINGARTEN SPACELIKE HYPERSURFACES IN LOCALLY SYMMETRIC LORENTZ SPACE

  • Yang, Dan
    • Bulletin of the Korean Mathematical Society
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    • v.49 no.2
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    • pp.271-284
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    • 2012
  • Let M be a linear Weingarten spacelike hypersurface in a locally symmetric Lorentz space with R = aH + b, where R and H are the normalized scalar curvature and the mean curvature, respectively. In this paper, we give some conditions for the complete hypersurface M to be totally umbilical.

A MAXIMUM PRINCIPLE FOR COMPLETE HYPERSURFACES IN LOCALLY SYMMETRIC RIEMANNIAN MANIFOLD

  • Zhang, Shicheng
    • Communications of the Korean Mathematical Society
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    • v.29 no.1
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    • pp.141-153
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    • 2014
  • In this article, we apply the weak maximum principle in order to obtain a suitable characterization of the complete linearWeingarten hypersurfaces immersed in locally symmetric Riemannian manifold $N^{n+1}$. Under the assumption that the mean curvature attains its maximum and supposing an appropriated restriction on the norm of the traceless part of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or hypersurface is an isoparametric hypersurface with two distinct principal curvatures one of which is simple.

LINEAR WEINGARTEN HYPERSURFACES IN RIEMANNIAN SPACE FORMS

  • Chao, Xiaoli;Wang, Peijun
    • Bulletin of the Korean Mathematical Society
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    • v.51 no.2
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    • pp.567-577
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    • 2014
  • In this note, we generalize the weak maximum principle in [4] to the case of complete linear Weingarten hypersurface in Riemannian space form $\mathbb{M}^{n+1}(c)$ (c = 1, 0,-1), and apply it to estimate the norm of the total umbilicity tensor. Furthermore, we will study the linear Weingarten hypersurface in $\mathbb{S}^{n+1}(1)$ with the aid of this weak maximum principle and extend the rigidity results in Li, Suh, Wei [13] and Shu [15] to the case of complete hypersurface.