• 호남수학학술지
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• 제31권3호
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• pp.381-398
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• 2009

Let $G\;=\;O(k){\times}O(k){\times}O(q)$ and let $M^n$ be a closed G-invariant minimal hypersurface with constant scalar curvature in $S^{n+1}$. Then we obtain a theorem: If $M^n$ has 2 distinct principal curvatures at some point p, then the square norm of the second fundamental form of $M^n$, S = n.

• East Asian mathematical journal
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• 제33권5호
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• pp.543-557
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• 2017

Jin [10] studied lightlike hypersurfaces of an indefinite trans-Sasakian manifold with a quarter-symmetric metric connection. We study further the geometry of this subject. The object of this paper is to study the geometry of half lightlike submanifolds of an indefinite trans-Sasakian manifold with a quarter-symmetric metric connection.

• 대한수학회논문집
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• 제12권1호
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• pp.109-112
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• 1997

In this paper, we study on a classification of hypersurfaces given by tansnormal functions on $R^{n+1}_1$. If M is a level set of a transnormal function on $R^{n+1}_1$, then it is one of a hyperplane, a cylinder around k-plane, a pseudo-sphere and a pseudo-hyperbolic space.

• 대한수학회보
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• 제49권2호
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• pp.285-293
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• 2012

In this paper, we investigate the hypersurface M in a unit sphere with constant k-th mean curvature and two distinct principal curvatures, and characterize such a hypersurface.

• 대한수학회보
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• 제44권1호
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• pp.157-172
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• 2007

In this paper, we prove that if the structure Jacobi operator $R_{\xi}-parallel\;and\;R_{\xi}$ commutes with the Ricci tensor S, then a real hypersurface with non-negative scalar curvature of a nonflat complex space form $M_{n}(C)$ is a Hopf hypersurface. Further, we characterize such Hopf hypersurface in $M_{n}(C)$.

• 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.

• 대한수학회지
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• 제48권2호
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• pp.253-266
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• 2011

In this paper we give an upper bound of the first eigenvalue of the Laplace operator on a complete stable minimal hypersurface M in the hyperbolic space which has finite $L^2$-norm of the second fundamental form on M. We provide some sufficient conditions for minimal hypersurface of the hyperbolic space to be stable. We also describe stability of catenoids and helicoids in the hyperbolic space. In particular, it is shown that there exists a family of stable higher-dimensional catenoids in the hyperbolic space.

• 대한수학회보
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• 제51권2호
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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.

• 대한수학회논문집
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• 제20권4호
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• pp.775-779
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• 2005

On a compact n-dimensional manifold $M^n$, a critical point of the total scalar curvature functional, restricted to the space of metrics with constant scalar curvature of volume 1, satifies the critical point equation (CPE), given by $Z_g\;=\;s_g^{1\ast}(f)$. It has been conjectured that a solution (g, f) of CPE is Einstein. The purpose of the present paper is to prove that every compact stable minimal hypersurface is in a certain hypersurface of $M^n$ under an assumption that Ker($s_g^{1\ast}{\neq}0$).

• 대한수학회논문집
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• 제17권2호
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• pp.261-278
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• 2002

Let G = O(2) $\times$ O(2) $\times$O(2) and let M$^4$be closed G-invariant minimal hypersurface with constant scalar curvature in S$^{5}$ . If M$^4$has 2 distinct principal curvatures at some point, then S = 4. Moreover, if S > 4, then M$^4$does not have simple principal curvatures everywhere.