• 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.

• 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.

• 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.

• Bulletin of the Korean Mathematical Society
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• v.60 no.5
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• pp.1299-1320
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• 2023

In the present paper, firstly we obtain the general expression of the canal hypersurfaces that are formed as the envelope of a family of pseudo hyperspheres, pseudo hyperbolic hyperspheres and null hyper-cones whose centers lie on a non-null curve with non-null Frenet vector fields in E⁴₁ and give their some geometric invariants such as unit normal vector fields, Gaussian curvatures, mean curvatures and principal curvatures. Also, we give some results about their flatness and minimality conditions and Weingarten canal hypersurfaces. Also, we obtain these characterizations for tubular hypersurfaces in E⁴₁ by taking constant radius function and finally, we construct some examples and visualize them with the aid of Mathematica.