• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.23 no.4
• /
• pp.351-365
• /
• 2019

This article presents a new approach for conformal flattening with optimal cone singularity. The algorithm here takes an optimal control for selecting optimal cones and uses the Ricci flow to force the flattening. This work is considered as a modification to the work of Soliman et al. [1] in the sense that they make use of the Yamabe equation for the flattening, which is an approximation of the Ricci flow. We present a numerical algorithm based on the optimal control with the mathematical background. Several numerical results validate that our method is optimal in total cone angle and usage of the Ricci flow ensures the conformal flattening while selecting optimal cones.

• Journal of the Korean Society for Industrial and Applied Mathematics
• /
• v.6 no.1
• /
• pp.59-65
• /
• 2002

In this paper, we consider some homogeneous maps from a cone over 2-spheres and determines whether they become energy minimizing maps or not. In fact, any homogeneous map from a standard cone over 2-sphere of radius smaller than 1 can not be a minimizing harmonic map.

• East Asian mathematical journal
• /
• v.38 no.3
• /
• pp.321-329
• /
• 2022

For an ample divisor A of birational type on a singular del Pezzo surface S of degree 4 with A₁-singularity, we show that the affine cone of S defined by A is flexible

• Journal of the Korean Mathematical Society
• /
• v.40 no.4
• /
• pp.709-725
• /
• 2003

We give an analytic approach to the versal deformation of a resolution of a germ of normal isolated singularities. In this paper, we treat only formal deformation theory and it is applied to complete the CR-description of the simultaneous resolution of a cone eve. a rational curve of degree n in P$^{n}$ (n $\leq$ 4).

• Journal of the Korean Mathematical Society
• /
• v.47 no.5
• /
• pp.967-983
• /
• 2010

Let M be an n-dimensional complete simply connected Riemannian manifold with sectional curvature bounded above by a nonpositive constant $-{\kappa}^2$. Using the cone total curvature TC($\Gamma$) of a graph $\Gamma$ which was introduced by Gulliver and Yamada [8], we prove that the density at any point of a soap film-like surface $\Sigma$ spanning a graph $\Gamma\;\subset\;M$ is less than or equal to $\frac{1}{2\pi}\{TC(\Gamma)-{\kappa}^2Area(p{\times}\Gamma)\}$. From this density estimate we obtain the regularity theorems for soap film-like surfaces spanning graphs with small total curvature. In particular, when n = 3, this density estimate implies that if $TC(\Gamma)$ < $3.649{\pi}\;+\;{\kappa}^2\inf\limits_{p{\in}F}Area(p{\times}{\Gamma})$, then the only possible singularities of a piecewise smooth (M, 0, $\delta$)-minimizing set $\Sigma$ are the Y-singularity cone. In a manifold with sectional curvature bounded above by $b^2$ and diameter bounded by $\pi$/b, we obtain similar results for any soap film-like surfaces spanning a graph with the corresponding bound on cone total curvature.