• 대한수학회논문집
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• 제39권3호
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• pp.739-756
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• 2024

The ordinary mean curvature vector field 𝗛 on a submanifold M of a space form is said to be proper if it satisfies equality Δ𝗛 = a𝗛 for a constant real number a. It is proven that every hypersurface of an Riemannian space form with proper mean curvature vector field has constant mean curvature. In this manuscript, we study the Lorentzian hypersurfaces with proper second mean curvature vector field of four dimensional Lorentzian space forms. We show that the scalar curvature of such a hypersurface has to be constant. In addition, as a classification result, we show that each Lorentzian hypersurface of a Lorentzian 4-space form with proper second mean curvature vector field is C-biharmonic, C-1-type or C-null-2-type. Also, we prove that every 𝗛₂-proper Lorentzian hypersurface with constant ordinary mean curvature in a Lorentz 4-space form is 1-minimal.

• 충청수학회지
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• 제34권4호
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• pp.369-378
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• 2021

In this paper, we give an upper bound for the fundamental tone of stable constant mean curvature hypersurfaces in hyperbolic space. Let M be an n-dimensional complete non-compact constant mean curvature hypersurface with finite L²-norm of the traceless second fundamental form. If M is weakly stable, then λ₁(M) is bounded above by n² + O(n^2+s) for arbitrary s > 0.

• 대한수학회보
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• 제48권2호
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• pp.335-342
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• 2011

In this paper, we prove that minimal hypersurfaces when n $\geq$ 3 and nonzero constant mean curvature hypersurfaces when n $\geq$ 2 foliated by spheres in parallel horizontal hyperplanes in $\mathbb{H}^n{\times}\mathbb{R}$ must be rotationally symmetric.

• 호남수학학술지
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• 제46권2호
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• pp.249-266
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• 2024

In this work, we study time-like and space-like surfaces invariant by a group of translation isometries of the half-space model ℋ³₁ of the anti-de Sitter space ℍ³₁ . We determine all such surfaces with constant mean curvature and constant Gaussian curvature. We also obtain umbilical surfaces of ℋ³₁.

• 대한수학회지
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• 제55권1호
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• pp.59-71
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• 2018

In the present paper, we study and classify factorable surfaces in a 3-dimensional isotropic space with constant isotropic Gaussian (K) and mean curvature (H). We provide a non-existence result relating to such surfaces satisfying ${\frac{H}{K}}=const$. Several examples are also illustrated.

• 대한수학회논문집
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• 제33권1호
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• pp.247-259
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• 2018

Factorable surfaces, i.e. graphs associated with the product of two functions of one variable, constitute a wide class of surfaces in differential geometry. Such surfaces in the pseudo-Galilean space with zero Gaussian and mean curvature were obtained in [2]. In this study, we provide new results relating to the factorable surfaces with non-zero constant Gaussian and mean curvature.

• 대한수학회지
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• 제38권1호
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• pp.135-146
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• 2001

Let M(sup)n be a space-ike submanifold in a de Sitter space M(sub)p(sup)n+p (c) with constant scalar curvature. We firstly extend Cheng-Yau's Technique to higher codimensional cases. Then we study the rigidity problem for M(sup)n with parallel normalized mean curvature vector field.

• 대한수학회보
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• 제54권1호
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• pp.159-175
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• 2017

We solve the $Bj{\ddot{o}}rling$ problem for constant mean curvature one surfaces in hyperbolic three-space and in de Sitter three-space. That is, we show that for any regular, analytic (and spacelike in the case of de Sitter three-space) curve ${\gamma}$ and an analytic (timelike in the case of de Sitter three-space) unit vector field N along and orthogonal to ${\gamma}$, there exists a unique (spacelike in the case of de Sitter three-space) surface of constant mean curvature 1 which contains ${\gamma}$ and the unit normal of which on ${\gamma}$ is N. Some of the consequences are the planar reflection principles, and a classification of rotationally invariant CMC 1 surfaces.

• 대한수학회지
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• 제31권3호
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• pp.339-352
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• 1994

A sumbanifold M of a quaternionic Kaehlerian manifold $\tilde{M}^m$ of real dimension 4m is called a generic submanifold if the normal space N(M) of M is always mapped into the tangent space T(M) under the action of the quaternionic Kaehlerian structure tensors of the ambient manifold at the same time.The purpose of the present paper is to study generic submanifold of quaternionic Kaehlerian manifold of constant Q-sectional curvature with nonvanishing parallel mean curvature vector. In section 1, we state general formulas on generic submanifolds of a quaternionic Kaehlerian manifold of constant Q-sectional curvature. Section 2 is devoted to the study generic submanifolds with nonvanishing parallel mean curvature vector and compute the restricted Laplacian for the second fundamental form in the direction of the mean curvature vector. As applications of those results, in section 3, we prove our main theorems. In this paper, the dimension of a manifold will always indicate its real dimension.

• 대한수학회보
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• 제42권1호
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• pp.39-44
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• 2005

Let M be a compact hypersurface in hyperbolic space and let A be the area of M and V be the volume of the compact domain bounded by M. In this paper, we find a lower bound for $\frac{A}{V}$ in two cases, M has constant scalar curvature and M has constant mean curvature.