• 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.

• Kyungpook Mathematical Journal
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• v.48 no.1
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• pp.109-121
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• 2008

We determine maximal space-like hypersurfaces which are the images of subbundles of the normal bundle of m-dimensional totally geodesic space-like submanifolds of an (m + 2)-dimensional Lorentzian space form $\tilde{M}_1^{m+2}$(c) under the normal exponential map. Then we construct examples of maximal space-like hypersurfaces of $\tilde{M}_1^{m+2}$(c).

• Journal of the Korean Mathematical Society
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• v.43 no.1
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• pp.147-157
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• 2006

In this paper, we study complete maximal space-like hypersurfaces with constant Gauss-Kronecker curvature in an antide Sitter space $H_1^4(-1)$. It is proved that complete maximal spacelike hypersurfaces with constant Gauss-Kronecker curvature in an anti-de Sitter space $H_1^4(-1)$ are isometric to the hyperbolic cylinder $H^2(c1){\times}H^1(c2)$ with S = 3 or they satisfy $S{\leq}2$, where S denotes the squared norm of the second fundamental form.

• Bulletin of the Korean Mathematical Society
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• v.47 no.5
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• pp.1067-1076
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• 2010

In this paper we investigate complete spacelike (r - 1)-maximal (i.e., $H_r\;{\equiv}\;0$) hypersurfaces with two distinct principal curvatures in the anti-de Sitter space $\mathbb{H}_1^{n+1}$(-1). We give a characterization of the hyperbolic cylinder.

• Kyungpook Mathematical Journal
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• v.31 no.2
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• pp.131-141
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• 1991

• East Asian mathematical journal
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• v.8 no.1
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• pp.25-39
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• 1992

• Bulletin of the Korean Mathematical Society
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• v.31 no.1
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• pp.85-92
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• 1994

It is well known that there exist no closed minimal surfaces in a 3-dimensional Euclidean space R$^{3}$. Myers [4] generalized the result to the case of the higher dimension and proved that there are no closed minimal hypersurfaces in an open hemisphere. The complete and non-compact version concerning Myers' theorem is recently considered by Cheng [1] and the following theorem is proved.

• Journal of the Korean Mathematical Society
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• v.53 no.4
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• pp.737-767
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• 2016

This paper concerns closed hypersurfaces of dimension $n{\geq}2$ in the hyperbolic space ${\mathbb{H}}_{\kappa}^{n+1}$ of constant sectional curvature evolving in direction of its normal vector, where the speed equals a power ${\beta}{\geq}1$ of the mean curvature. The main result is that if the initial closed, weakly h-convex hypersurface satisfies that the ratio of the biggest and smallest principal curvature at everywhere is close enough to 1, depending only on n and ${\beta}$, then under the flow this is maintained, there exists a unique, smooth solution of the flow which converges to a single point in ${\mathbb{H}}_{\kappa}^{n+1}$ in a maximal finite time, and when rescaling appropriately, the evolving hypersurfaces exponential convergence to a unit geodesic sphere of ${\mathbb{H}}_{\kappa}^{n+1}$.

• Bulletin of the Korean Mathematical Society
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• v.58 no.4
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• pp.877-884
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• 2021

We prove that if a compact convex hypersurface of ℝⁿ⁺¹ has almost maximal extinction time when it is evolved by the mean curvature flow, then it must be nearly round in the C⁰-norm.

• Journal of the Korean Mathematical Society
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• v.50 no.6
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• pp.1311-1332
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• 2013

This paper concerns the evolution of a closed hypersurface of the hyperbolic space, convex by horospheres, in direction of its inner unit normal vector, where the speed equals a positive power ${\beta}$ of the positive mean curvature. It is shown that the flow exists on a finite maximal interval, convexity by horospheres is preserved and the hypersurfaces shrink down to a single point as the final time is approached.