• Korean Journal of Mathematics
• /
• v.24 no.4
• /
• pp.613-626
• /
• 2016

In this article, we study skew ruled surfaces by using the geodesic Frenet trihedron of its generator. We obtained some conditions on this surface to ensure that this ruled surface is flat, II-flat, minimal, II-minimal and Weingarten surface. Moreover, the parametric equations of asymptotic and geodesic lines on this ruled surface are determined and illustrated through example using the program of mathematica.

• Honam Mathematical Journal
• /
• v.30 no.4
• /
• pp.677-683
• /
• 2008

On a compact n-dimensional manifold M, it has been conjectured that a critical point metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of volume 1, should be Einstein. The purpose of the present paper is to prove that a 3-dimensional manifold (M,g) is isometric to a standard sphere if ker $s^*_g{{\neq}}0$ and there is a lower Ricci curvature bound. We also study the structure of a compact oriented stable minimal surface in M.

• Bulletin of the Korean Mathematical Society
• /
• v.58 no.4
• /
• pp.959-972
• /
• 2021

In this paper, we define and classify minimal translation surfaces with respect to a kind of semi-symmetric metric connections and a kind of semi-symmetric non-metric connections in ℝ³ and ℝ³₁.

• Journal of the Chungcheong Mathematical Society
• /
• v.34 no.3
• /
• pp.307-316
• /
• 2021

We provide a gap theorem of Simons' type for free boundary minimal and constant mean curvature surfaces in the unit ball in 3-dimensional Euclidean space.

• The Korean Journal of Physiology
• /
• v.5 no.2
• /
• pp.63-70
• /
• 1971

In an attempt to understand the possible effects of temperature and X-irradiation on the activities of surfactant in rabbits, the pulmonary surfactant from the rabbit was subjected to the incubation at $37^{\circ}C$ and X-irradiation with 900r in vitro, and activities of surfactant were measured at 10, 30, 60, and 90 minutes. Tension-area diagram of the lung extract was recorded automatically by the modified Langmuir-wilhelmy balance with a synchronized recording system designed in this Department. A comparison was made with the normal and the following results were obtained. 1) The maximal surface tension, minimal surface tension, width of the tension area diagram at the surface area of 40% and stability index of the normal rabbit lung extract were $31.6{\pm}3.11\;dynes/cm,\;8.2{\pm}0.56\;dynes/cm,\;21.4{\pm}4.40\;dynes/cm\;and\;1.12{\pm}0.22$,respectively. 2) In the $37^{\circ}C$ incubation group, maximal surface tension was similar to the normal value, while minimal surface tension was significantly lower and stability infer was markedly higher than the normal. 3) In the group where X-irradiation of 900r in vitro was applied, maximal surface tension did not differ greatly with the normal or the $37^{\circ}C$ incubation group. The minimal surface tension was significantly lower than the normal but comparing with the $37^{\circ}C$ incubation group, some decrease in minimal surface tension was noted. The width of the tension·area diagram at 40% and stability index in the irradiated group were significantly higher than the normal but a tendency of increase was noted comparing with the $37^{\circ}C$ incubation group.

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

• Journal of the Korean Mathematical Society
• /
• v.56 no.5
• /
• pp.1173-1246
• /
• 2019

We find all P-resolutions of quotient surface singularities (especially, tetrahedral, octahedral, and icosahedral singularities) together with their dual graphs, which reproduces (a corrected version of) Jan Steven's list [Manuscripta Math. 1993] of the numbers of P-resolutions of each singularities. We then compute the dimensions and Milnor numbers of the corresponding irreducible components of the reduced base spaces of versal deformations of each singularities. Furthermore we realize Milnor fibers as complements of certain divisors (depending only on the singularities) in rational surfaces via the semi-stable minimal model program for 3-folds. Then we compare Milnor fibers with minimal symplectic fillings, where the latter are classified by Bhupal and Ono [Nagoya Math. J. 2012]. As an application, we show that there are 6 pairs of entries in the list of Bhupal and Ono [Nagoya Math. J. 2012] such that two entries in each pairs represent diffeomorphic minimal symplectic fillings.

• East Asian mathematical journal
• /
• v.40 no.1
• /
• pp.75-83
• /
• 2024

For a reduced, irreducible and nondegenerate curve C ⊂ ℙ^r of degree d, it was shown that the arithmetic genus g of C has an upper bound π₀(d, r) by G. Castelnuovo. And he also classified the curves that attain the extremal value. These curves are arithmetically Cohen-Macaulay and contained in a surface of minimal degree. In this paper, we investigate the arithmetic genus of curves lie on a surface of minimal degree - the Veronese surface, smooth rational normal surface scrolls and singular rational normal surface scrolls. We also provide a construction of curves on singular rational normal surface scroll S(0, 2) ⊂ ℙ³ which attain the maximal arithmetic genus.

• Journal of Institute of Convergence Technology
• /
• v.10 no.1
• /
• pp.7-12
• /
• 2020

We describe a efficient surface reconstruction method that reconstructs a 3D manifold polygonal mesh approximately passing through a set of 3D oriented points. Our algorithm includes 3D convex hull, octree data structure, signed distance function (SDF), and marching cubes. The 3D convex hull provides us with a fast computation of SDF, octree structure allows us to compute a minimal distance for SDF, and marching cubes lead to iso-surface generation with SDF. Our approach gives us flexibility in the choice of the resolution of the reconstructed surface, and it also enables to use on low-level PCs with minimal peak memory usage. Experimenting with publicly available scan data shows that we can reconstruct a polygonal mesh from point cloud of sizes varying from 10,000 ~ 1,000,000 in about 1~60 seconds.

• Bulletin of the Korean Mathematical Society
• /
• v.50 no.4
• /
• pp.1329-1343
• /
• 2013

In this paper, we define translation surfaces in the 3-dimensional Heisenberg group $\mathcal{H}_3$ obtained as a product of two planar curves lying in planes, which are not orthogonal, and study minimal translation surfaces in $\mathcal{H}_3$.