• Journal of the Korean Mathematical Society
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• v.54 no.1
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• pp.303-317
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• 2017

An isometry of hyperbolic space can be written as a composition of the reflection in the isometric sphere and two Euclidean isometries on the boundary at infinity. The isometric sphere is also used to construct the Ford fundamental domains for the action of discrete groups of isometries. In this paper, we study the isometric spheres of isometries acting on hyperbolic 4-space. This is a new phenomenon which occurs in hyperbolic 4-space that the two isometric spheres of a parabolic isometry can intersect transversally. We provide one geometric way to classify isometries of hyperbolic 4-space using the isometric spheres.

• Bulletin of the Korean Mathematical Society
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• v.59 no.2
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• pp.351-360
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• 2022

We prove that a punctured-torus group of hyperbolic 4-space which keeps an embedded hyperbolic 2-plane invariant has a strictly parabolic commutator. More generally, this rigidity persists for a punctured-surface group.

• 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.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.

• Honam Mathematical Journal
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• v.29 no.3
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• pp.481-493
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• 2007

We will construct several types of semi-regular tilings of a hyperbolic unit disk model by defining geometric features of the definition of distance in a hyperbolic plane, area of triangle, and isometry of inversions. We researched the method of regular tilings and semi-regular tilings of hyperbolic unit disk model and wrote an semi-regular tiling construction algorithm using Cabri2 program and Cinderella program. Lastly, We want to make a product related to traditional heritage cultural patterns using Photoshop, so we'll model the advertising photos of cites; Seoul, Gwangju.

• Journal of the Chungcheong Mathematical Society
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• v.22 no.4
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• pp.771-776
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• 2009

Let (H, g) denote the upper half plane in $R^2$ with the Riemannian metric g := ($(dx)^2$ + $(dy)^2$)$/y^2$. First of all we get a necessary and sufficient condition for a diffeomorphism $\phi$ of (H, g) to be a harmonic map. And, we obtain the fact that if a diffeomorphism $\phi$ of (H, g) is a harmonic function, then the following facts are equivalent: (1) $\phi$ is a harmonic map; (2) $\phi$ is an affine transformation; (3) $\phi$ is an isometry (motion).

• Journal of the Korean Mathematical Society
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• v.48 no.1
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• pp.63-82
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• 2011

On the hyperbolic space $D^n$, codimension-one totally geodesic foliations of class $C^k$ are classified. Except for the unique parabolic homogeneous foliation, the set of all such foliations is in one-one correspondence (up to isometry) with the set of all functions z : [0, $\pi$] $\rightarrow$ $S^{n-1}$ of class $C^{k-1}$ with z(0) = $e_1$ = z($\pi$) satisfying |z'(r)| ${\leq}1$ for all r, modulo an isometric action by O(n-1) ${\times}\mathbb{R}{\times}\mathbb{Z}_2$. Since 1-dimensional metric foliations on $D^n$ are always either homogeneous or flat (that is, their orthogonal distributions are integrable), this classifies all 1-dimensional metric foliations as well. Equations of leaves for a non-trivial family of metric foliations on $D^2$ (called "fifth-line") are found.

• Communications of the Korean Mathematical Society
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• v.22 no.2
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• pp.277-287
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• 2007

We prove that if a graph G of bounded degree has finitely many p-hyperbolic ends($1) in which every bounded energy finite p-harmonic function is asymptotically constant for almost every path, then the set $\mathcal{HBD}_p(G)$ of all bounded energy finite p-harmonic functions on G is in one to one corresponding to $\mathbf{R}^l$, where $l$ is the number of p-hyperbolic ends of G. Furthermore, we prove that if a graph G' is roughly isometric to G, then $\mathcal{HBD}_p(G')$ is also in an one to one correspondence with $\mathbf{R}^l$.