• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1795-1804
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• 2016

Let $M^n$, $2{\leq}n{\leq}6$ be a complete noncompact hypersurface immersed in ${\mathbb{H}}^{n+1}$. We show that there exist two certain positive constants 0 < ${\delta}{\leq}1$, and ${\beta}$ depending only on ${\delta}$ and the first eigenvalue ${\lambda}_1(M)$ of Laplacian such that if M satisfies a (${\delta}$-SC) condition and ${\lambda}_1(M)$ has a lower bound then $H^1(L^2(M))=0$. Excepting these two conditions, there is no more additional condition on the curvature.

• 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.21 no.1
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• pp.38-40
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• 1984

• East Asian mathematical journal
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• v.11 no.2
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• pp.327-340
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• 1995

• East Asian mathematical journal
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• v.11 no.1
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• pp.87-113
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• 1995

• Journal of the Korean Mathematical Society
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• v.36 no.3
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• pp.581-592
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• 1999

Let M be a properly immersed timelike hypersurface of $\overline{M}$. If M is a diagonal type, M satisfies the generic condition under the certain conditions of the eigenvalues of the shape operator. Moreover, applying them to Raychaudhuri equation, we can show that M satisfies the generic condition. Thus, by these results, we establish the singularity theorem for M in $\overline{M}$.

• Communications of the Korean Mathematical Society
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• v.37 no.3
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• pp.893-904
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• 2022

In this note, we apply a maximum principle related to volume growth of a complete noncompact Riemannian manifold, which was recently obtained by Alías, Caminha and do Nascimento in [4], to establish new uniqueness and nonexistence results concerning maximal spacelike hypersurfaces immersed in a generalized Robertson-Walker (GRW) spacetime obeying the timelike convergence condition. A study of entire solutions for the maximal hypersurface equation in GRW spacetimes is also made and, in particular, a new Calabi-Bernstein type result is presented.

• Bulletin of the Korean Mathematical Society
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• v.55 no.1
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• pp.227-237
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• 2018

In this paper, we study some geometric structures of a weakly cyclic Z-symmetric manifold (briefly, $[W CZS]_n$). More precisely, we prove that a conformally flat $[W CZS]_n$ satisfying certain conditions is special conformally flat and hence the manifold can be isometrically immersed in an Euclidean manifold $E^n+1$ as a hypersurface if the manifold is simply connected. Also we show that there exists a $[W CZS]_4$ with one parameter family of its associated 1-forms.

• Bulletin of the Korean Mathematical Society
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• v.50 no.1
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• pp.97-103
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• 2013

As a suitable application of the well known generalized maximum principle of Omori-Yau, we obtain rigidity results concerning to a complete hypersurface immersed with bounded mean curvature in the $(n+1)$-dimensional hyperbolic space $\mathbb{H}^{n+1}$. In our approach, we explore the existence of a natural duality between $\mathbb{H}^{n+1}$ and the half $\mathcal{H}^{n+1}$ of the de Sitter space $\mathbb{S}_1^{n+1}$, which models the so-called steady state space.

• Communications of the Korean Mathematical Society
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• v.30 no.2
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• pp.131-142
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• 2015

The purpose of this paper is to study real hypersurfaces immersed in a complex hyperbolic space $CH^n$ and especially to investigate certain curvature conditions for such real hypersurfaces to be the model hypersurfaces in classification theorem (said to be Theorem M-R) given by Montiel and Romero ([4]) in Section 3.