• Journal of the Korean Mathematical Society
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• v.46 no.3
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• pp.513-521
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• 2009

We first introduce a complex hyperbolic space and a complex hyperbolic manifold. After defining the canonical horoball and the canonical cusp on the complex hyperbolic manifold, we estimate the volumes of canonical cusps of complex hyperbolic manifolds. Finally, we deal with cusped, complex hyperbolic 2-manifolds, and in particular, the ones with only one cusp.

• Bulletin of the Korean Mathematical Society
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• v.60 no.3
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• pp.717-732
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• 2023

The goal of this paper is to analyze the generalized m-quasi-Einstein structure in the context of almost Kenmotsu manifolds. Firstly we showed that a complete Kenmotsu manifold admitting a generalized m-quasi-Einstein structure (g, f, m, λ) is locally isometric to a hyperbolic space ℍ²ⁿ⁺¹(-1) or a warped product ${\tilde{M}}{\times}{_{\gamma}{\mathbb{R}}$ under certain conditions. Next, we proved that a (κ, µ)'-almost Kenmotsu manifold with h' ≠ 0 admitting a closed generalized m-quasi-Einstein metric is locally isometric to some warped product spaces. Finally, a generalized m-quasi-Einstein metric (g, f, m, λ) in almost Kenmotsu 3-H-manifold is considered and proved that either it is locally isometric to the hyperbolic space ℍ³(-1) or the Riemannian product ℍ²(-4) × ℝ.

• Journal of the Korean Mathematical Society
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• v.50 no.2
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• pp.425-444
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• 2013

We construct some hyperbolic hyperelliptic space forms whose fundamental groups are generated by only two or three isometries. Each occurring group is obtained from a supergroup, which is an extended Coxeter group generated by plane re ections and half-turns. Then we describe covering properties and determine the isometry groups of the constructed manifolds. Furthermore, we give an explicit construction of space form of the second smallest volume nonorientable hyperbolic 3-manifold with one cusp.

• Journal of the Korean Mathematical Society
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• v.49 no.2
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• pp.343-356
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• 2012

In this paper, we consider the canonical cusps in complex hyperbolic surfaces. We will classify canonical cusps in complex hyper-bolic surfaces and find correspondence between them and 3-dimensiona nilpotent groups. This paper is a sequel of our paper [6].

• Journal of the Korean Mathematical Society
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• v.44 no.5
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• pp.1093-1102
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• 2007

In this note we survey some recent results on symmetric space, and related topics like rigidity, flexibility and geometry-topology aspect of symmetric space, specially in the line of hyperbolic 3-manifolds.

• Communications of the Korean Mathematical Society
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• v.28 no.3
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• pp.581-587
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• 2013

Let ${\Lambda}$ be a robustly transitive set of a diffeomorphism $f$ on a closed $C^{\infty}$ manifold. In this paper, we characterize hyperbolicity of ${\Lambda}$ in $C^1$-generic sense.

• Journal of the Korean Mathematical Society
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• v.40 no.4
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• pp.595-607
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• 2003

In the previous paper [6], the author constructed hyperbolic hypersurfaces of degree $2^{n}$ in the n-dimensional complex projective space for every $n\;\geq\;3$. The purpose of this paper is to give some improvement of this result and to show some general methods of constructions of hyperbolic hypersurfaces of higher degree, which enable us to construct hyperbolic hypersurfaces of degree d in the n-dimensional complex projective space for every $d\;\geq\;2\;{\times}\;6^{n}$.

• Journal of the Korean Mathematical Society
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• v.45 no.4
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• pp.1007-1025
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• 2008

We will describe a way to construct Ford domains of cusped hyperbolic 3-manifolds on maximal cusp diagrams and compute fundamental groups using face pairing maps as generators and Cannon-Floyd-Parry's edge cycles as relations. We also describe explicitly a cutting and pasting alteration to reduce the number of faces on the bottom region of Ford domains. We expect that our analysis of Ford domains will be useful on other future research.

• Journal of the Korean Mathematical Society
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• v.61 no.3
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• pp.495-513
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• 2024

In the paper, we show that for a generic C¹ vector field X on a closed three dimensional manifold M, any isolated transitive set of X is singular hyperbolic. It is a partial answer of the conjecture in [13].

• Journal of the Korean Mathematical Society
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• v.58 no.3
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• pp.597-607
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• 2021

The present paper deals with the study of Fischer-Marsden conjecture on a Kenmotsu manifold. It is proved that if a Kenmotsu metric satisfies 𝔏^*_g(λ) = 0 on a (2n + 1)-dimensional Kenmotsu manifold M²ⁿ⁺¹, then either ξλ = -λ or M²ⁿ⁺¹ is Einstein. If n = 1, M³ is locally isometric to the hyperbolic space H³ (-1).