• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.2081-2089
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• 2017

In this paper we generalize 3-dimensional Bour's Theorem to the case of 4-dimension. We proved that a helicoidal surface in $\mathbb{R}^4$ is isometric to a family of surfaces of revolution in $\mathbb{R}^4$ in such a way that helices on the helicoidal surface correspond to parallel circles on the surfaces of revolution. Moreover, if the surfaces are required further to have the same Gauss map, then they are hyperplanar and minimal. Parametrizations for such minimal surfaces are given explicitly.

• Honam Mathematical Journal
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• v.34 no.2
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• pp.191-198
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• 2012

Minimal surfaces with given boundaries are the solutions of Plateau's problem. In studying the calculus of variations for the minimal surfaces, the functional ${\varepsilon}$, corresponding to the energy of surfaces, is introduced in [Ki09]. In this paper we derive a formula for the second derivative of ${\varepsilon}$, which is necessary for further theories of the calculus of variations.

• Bulletin of the Korean Mathematical Society
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• v.53 no.2
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• pp.531-540
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• 2016

In this work, we introduce the complete Riemannian manifold $\mathbb{F}_3$ which is a three-dimensional real vector space endowed with a conformally flat metric that is a solution of the Einstein equation. We obtain a second order nonlinear ordinary differential equation that characterizes the helicoidal minimal surfaces in $\mathbb{F}_3$. We show that the helicoid is a complete minimal surface in $\mathbb{F}_3$. Moreover we obtain a local solution of this differential equation which is a two-parameter family of functions ${\lambda}_h,K_2$ explicitly given by an integral and defined on an open interval. Consequently, we show that the helicoidal motion applied on the curve defined from ${\lambda}_h,K_2$ gives a two-parameter family of helicoidal minimal surfaces in $\mathbb{F}_3$.

• Journal for History of Mathematics
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• v.23 no.4
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• pp.31-46
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• 2010

Theory of minimal surfaces has always been in the center of differential geometry. The most difficult part in minimal surfaces is how to find meaningful examples. In this paper we survey the history of search for minimal surfaces. We also introduce examples of recently emerging maximal surfaces in the Lorentz-Minkowski space and compare the processes in the search for the minimal and the maximal surfaces.

• Bulletin of the Korean Mathematical Society
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• v.56 no.6
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• pp.1539-1550
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• 2019

We classify minimal surfaces in ${\mathbb{S}}^3$ which are foliated by circles and ruled constant mean curvature (cmc) surfaces in ${\mathbb{S}}^3$. First we show that minimal surfaces in ${\mathbb{S}}^3$ which are foliated by circles are either ruled (that is, foliated by geodesics) or rotationally symmetric (that is, invariant under an isometric ${\mathbb{S}}^1$-action which fixes a geodesic). Secondly, we show that, locally, there is only one ruled cmc surface in ${\mathbb{S}}^3$ up to isometry for each nonnegative mean curvature. We give a parametrization of the ruled cmc surface in ${\mathbb{S}}^3$(cf. Theorem 3).

• Bulletin of the Korean Mathematical Society
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• v.52 no.5
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• pp.1433-1443
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• 2015

Catenoid and Riemann's minimal surface are foliated by circles, that is, they are union of circles. In $\mathbb{R}^3$, there is no other nonplanar example of circle-foliated minimal surfaces. In $\mathbb{R}^4$, the graph $G_c$ of w = c/z for real constant c and ${\zeta}{\in}\mathbb{C}{\backslash}\{0}$ is also foliated by circles. In this paper, we show that every circle-foliated minimal surface in $\mathbb{R}$ is either a catenoid or Riemann's minimal surface in some 3-dimensional Affine subspace or a graph surface $G_c$ in some 4-dimensional Affine subspace. We use the property that $G_c$ is circle-foliated to construct circle-foliated minimal surfaces in $S^4$ and $H^4$.

• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1887-1892
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• 2016

It is well known that the helicoids are the only ruled minimal surfaces in ${\mathbb{R}}^3$. The similar characterization for ruled minimal surfaces can be given in many other 3-dimensional homogeneous spaces. In this note we consider the product space $M{\times}{\mathbb{R}}$ for a 2-dimensional manifold M and prove that $M{\times}{\mathbb{R}}$ has a nontrivial minimal surface ruled by horizontal geodesics only when M has a Clairaut parametrization. Moreover such minimal surface is the trace of the longitude rotating in M while translating vertically in constant speed in the direction of ${\mathbb{R}}$.

• Honam Mathematical Journal
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• v.27 no.3
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• pp.465-476
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• 2005

In this article, we solve some kinds of non-compact Douglas-Plateau problem for two convex curves in parallel planes.

• Journal of the Korean Mathematical Society
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• v.55 no.5
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• pp.1235-1255
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• 2018

We provide definitions for the helicoidal Killing field and the helicoid in arbitrary three-manifolds, and investigate helicoids and ruled minimal surfaces in homogeneous three-manifolds, mainly in $SL_2{\mathbb{R}}$ and Sol(3). In so doing we finish our classification of ruled minimal surfaces in homogeneous three-manifolds with the isometry group of dimension 4.

• Communications of the Korean Mathematical Society
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• v.23 no.4
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• pp.587-595
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• 2008

It is well known that critical points of the total scalar curvature functional S on the space of all smooth Riemannian structures of volume 1 on a compact manifold M are exactly the Einstein metrics. When the domain of S is restricted to the space of constant scalar curvature metrics, there has been a conjecture that a critical point is isometric to a standard sphere. In this paper we investigate the relationship between the first Betti number and stable minimal surfaces, and study the analytic properties of stable minimal surfaces in M for n = 3.