• Bulletin of the Korean Mathematical Society
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• v.35 no.2
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• pp.301-309
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• 1998

The purpose of this paper is to give a theorem of Liouvilletype for complete Riemannian manifolds as an extension of the Theorem of Nishikawa [6].

• 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.49 no.3
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• pp.647-653
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• 2012

In this paper, we obtain a generalized Lorentzian splitting theorem by weakening the assumption of nonnegative Ricci curvature.

• The Pure and Applied Mathematics
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• v.23 no.1
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• pp.13-19
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• 2016

In this paper, we study some characterizations of unbounded domains. Among these, so-called G-domain is introduced by Cabre for the Aleksandrov-Bakelman-Pucci maximum principle of second order linear elliptic operator in a non-divergence form. This domain is generalized to wG-domain by Vitolo for the maximum principle of an unbounded domain, which contains G-domain. We study the properties of these domains and compare some other characterizations. We prove that sA-domain is wG-domain, but using the Cantor set, we are able to construct a example which is wG-domain but not sA-domain.

• Journal of the Korean Mathematical Society
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• v.54 no.3
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• pp.763-772
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• 2017

In this paper, we study the Liouville type theorems for transversally harmonic and biharmonic maps on foliated Riemannian manifolds.

• 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.

• 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.

• Korean Journal of Mathematics
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• v.23 no.2
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• pp.283-291
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• 2015

In the optimal control problem, at first we search the expected optimal solution by using Pontryagin type's necessary conditions called the maximum principle. Next we use the sufficient conditions to conclude that the searched solution is optimal. In this article the sufficient conditions are studied. The value function is used for sufficient conditions.

• Bulletin of the Korean Mathematical Society
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• v.58 no.5
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• pp.1053-1068
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• 2021

We investigate the spacelike hypersurface Mⁿ with constant mean curvature (CMC) in a Lorentz Einstein manifold Lⁿ⁺¹₁, which is supposed to obey some appropriate curvature constraints. Applying a suitable Simons type formula jointly with the well known generalized maximum principle of Omori-Yau, we obtain some rigidity classification theorems and pinching theorems of hypersurfaces.

• Korean Journal of Mathematics
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• v.17 no.2
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• pp.127-145
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• 2009

In this paper we introduce a new subclass $K_{\mu}^{\lambda},{\phi},{\eta}(n;{\rho};{\alpha})$ of analytic and multivalent functions with negative coefficients using fractional calculus operators. Connections to the well known and some new subclasses are discussed. A necessary and sufficient condition for a function to be in $K_{\mu}^{\lambda},{\phi},{\eta}(n;{\rho};{\alpha})$ is obtained. Several distortion inequalities involving fractional integral and fractional derivative operators are also presented. We also give results for radius of starlikeness, convexity and close-to-convexity and inclusion property for functions in the subclass. Modified Hadamard product, application of class preserving integral operator and other interesting properties are also discussed.