• Journal of the Korean Mathematical Society
• /
• v.54 no.2
• /
• pp.545-561
• /
• 2017

The signature of a surface bundle over a surface is known to be divisible by 4. It is also known that the signature vanishes if the fiber genus ${\leq}2$ or the base genus ${\leq}1$. In this article, we construct new smooth 4-manifolds with signature 4 which are surface bundles over surfaces with small fiber and base genera. From these we derive improved upper bounds for the minimal genus of surfaces representing the second homology classes of a mapping class group.

• Journal of the Korean Mathematical Society
• /
• v.58 no.1
• /
• pp.109-122
• /
• 2021

In this paper, we consider the problem of finding the hypersurface Mn in the Euclidean (n + 1)-space ℝn+1 that satisfies an equation of mean curvature type, called singular minimal hypersurface equation. Such an equation physically characterizes the surfaces in the upper half-space ℝ+3 (u) with lowest gravity center, for a fixed unit vector u ∈ ℝ3. We first state that a singular minimal cylinder Mn in ℝn+1 is either a hyperplane or a α-catenary cylinder. It is also shown that this result remains true when Mn is a translation hypersurface and u is a horizantal vector. As a further application, we prove that a singular minimal translation graph in ℝ3 of the form z = f(x) + g(y + cx), c ∈ ℝ - {0}, with respect to a certain horizantal vector u is either a plane or a α-catenary cylinder.

• Communications of the Korean Mathematical Society
• /
• v.28 no.3
• /
• pp.425-432
• /
• 2013

Let S be a minimal surface of general type with $p_g(S)=q(S)=0$ having an involution ${\sigma}$ over the field of complex numbers. It is well known that if the bicanonical map ${\varphi}$ of S is composed with ${\sigma}$, then the minimal resolution W of the quotient $S/{\sigma}$ is rational or birational to an Enriques surface. In this paper we prove that the surface W of S with $K^2_S=5,6,7,8$ having an involution ${\sigma}$ with which the bicanonical map ${\varphi}$ of S is composed is rational. This result applies in part to surfaces S with $K^2_S=5$ for which ${\varphi}$ has degree 4 and is composed with an involution ${\sigma}$. Also we list the examples available in the literature for the given $K^2_S$ and the degree of ${\varphi}$.

• Journal of the Korean Society for Precision Engineering
• /
• v.28 no.7
• /
• pp.834-850
• /
• 2011

In this paper, a novel tissue engineering scaffold design method based on triply periodic minimal surface (TPMS) is proposed. After generating the hexahedral elements for a 3D anatomical shape using the distance field algorithm, the unit cell libraries composed of triply periodic minimal surfaces are mapped into the subdivided hexahedral elements using the shape function widely used in the finite element method. In addition, a heterogeneous implicit solid representation method is introduced to design a 3D (Three-dimensional) bio-mimetic scaffold for tissue engineering from a sequence of computed tomography (CT) medical image data. CT image of a human spine bone is used as the case study for designing a 3D bio-mimetic scaffold model from CT image data.

• Bulletin of the Korean Mathematical Society
• /
• v.31 no.1
• /
• pp.85-92
• /
• 1994

It is well known that there exist no closed minimal surfaces in a 3-dimensional Euclidean space R$^{3}$. Myers [4] generalized the result to the case of the higher dimension and proved that there are no closed minimal hypersurfaces in an open hemisphere. The complete and non-compact version concerning Myers' theorem is recently considered by Cheng [1] and the following theorem is proved.

• Kyungpook Mathematical Journal
• /
• v.49 no.2
• /
• pp.321-348
• /
• 2009

In this paper, we give two lower bounds on the diameter of the boundary slope set of a Montesinos knot. One is described in terms of the minimal crossing numbers of the knots, and the other is related to the Euler characteristics of essential surfaces with the maximal/minimal boundary slopes.

• Honam Mathematical Journal
• /
• v.38 no.1
• /
• pp.151-167
• /
• 2016

In Minkowskian 3-spacetime $L^3$ we find timelike or spacelike surface of revolution for the given Gauss curvature K = -1, 0, 1 and mean curvature H = 0. In fact, we set up the surface of revolution with the time axis for z-axis to be able to draw those surfaces on standard pictures in Minkowskian 3-spacetime $L^3$.

• Communications of the Korean Mathematical Society
• /
• v.32 no.3
• /
• pp.479-494
• /
• 2017

We construct various examples of simply connected minimal complex surfaces of general type with $p_g=0$ and $K^2=1,2$ using ${\mathbb{Q}}$-Gorenstein smoothing method.

• Journal of the Korean Mathematical Society
• /
• v.50 no.3
• /
• pp.493-507
• /
• 2013

We construct a new family of simply connected minimal complex surfaces of general type with $p_g$ = 1, $q$ = 0, and $K^2$ = 3, 4, 5, 6, 8 using a $\mathbb{Q}$-Gorenstein smoothing theory.

• Journal of the Korean Mathematical Society
• /
• v.44 no.2
• /
• pp.407-442
• /
• 2007

The study of Euclidean submanifolds with finite type "classical" Gauss map was initiated by B.-Y. Chen and P. Piccinni in [11]. On the other hand, it was believed that for spherical sub manifolds the concept of spherical Gauss map is more relevant than the classical one (see [20]). Thus the purpose of this article is to initiate the study of spherical submanifolds with finite type spherical Gauss map. We obtain several fundamental results in this respect. In particular, spherical submanifolds with 1-type spherical Gauss map are classified. From which we conclude that all isoparametric hypersurfaces of $S^{n+1}$ have 1-type spherical Gauss map. Among others, we also prove that Veronese surface and equilateral minimal torus are the only minimal spherical surfaces with 2-type spherical Gauss map.