• Bulletin of the Korean Mathematical Society
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• v.49 no.3
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• pp.581-588
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• 2012

We find a $C^{\infty}$ one-parameter family of Riemannian metrics $g_t$ on $\mathbb{R}^3$ for $0{\leq}t{\leq}{\varepsilon}$ for some number ${\varepsilon}$ with the following property: $g_0$ is the Euclidean metric on $\mathbb{R}^3$, the scalar curvatures of $g_t$ are strictly decreasing in t in the open unit ball and $g_t$ is isometric to the Euclidean metric in the complement of the ball.

• East Asian mathematical journal
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• v.15 no.1
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• pp.81-90
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• 1999

• Journal of Korea Multimedia Society
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• v.6 no.4
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• pp.698-706
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• 2003

The primary goal of surface approximation is to reduce the degree of deviation of the simplified surface from the original surface. However it is difficult to define the metric that can measure the amount of deviation quantitatively. Many of the existing studies analogize it by using the change of the scalar quantity before and after simplification. This approach makes a lot of sense in the point that the local surfaces with small scalar are relatively less important since they make a low impact on the adjacent areas and thus can be removed from the current surface. However using scalar value alone there can exist many cases that cannot compute the degree of geometric importance of local surface. Especially the perceptual geometric features providing important clues to understand an object, in our observation, are generally constructed with small scalar value. This means that the distinguishing features can be removed in the earlier stage of the simplification process. In this paper, to resolve this problem, we present various factors and their combination as the metric for calculating the deviation error by introducing the orientation of local surfaces. Experimental results indicate that the surface orientation has an important influence on measuring deviation error and the proposed combined error metric works well retaining the relatively high curvature regions on the object's surface constructed with various and complex curvatures.

• Journal of the Korean Mathematical Society
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• v.34 no.4
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• pp.1037-1048
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• 1997

We show that an asymptotically Euclidean scalar-flat Kahler metric ona complex surface can be smoothly conformally compactified at the infinity point. We discuss some implication of this, characterizing such metrics on $C^2$ blown up at some points.

• Bulletin of the Korean Mathematical Society
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• v.61 no.1
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• pp.281-290
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• 2024

We consider the total scalar curvature functional, and show that if the second variation in the transverse traceless tensor direction is negative, then the metric is Einstein. We also find the relation between the second variation and the Lichnerowicz Laplacian.

• The Pure and Applied Mathematics
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• v.4 no.1
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• pp.27-33
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• 1997

Let (M = B$\times$f F, g) be an ($n \geq3$ )-dimensional differential manifold with Riemannian metric g. We solve the following elliptic nonlinear partial differential equation (equation omitted). where $\Delta_{g}$ is the Laplacian in the $\Delta$g-metric and ($h(\chi)$) is the scalar curvature of g and ($H(\chi)$) is a function on M.

• Bulletin of the Korean Mathematical Society
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• v.35 no.3
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• pp.421-431
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• 1998

We characterize real 4-dimensional Kahler metrices which are critical for natural quadratic Riemannian functionals defined on the space of all Riemannian metrics. In particular we show that such critical Kahler surfaces are either Einstein or have zero scalar curvature. We also make some discussion on criticality in the space of Kahler metrics.

• Kyungpook Mathematical Journal
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• v.62 no.3
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• pp.485-495
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• 2022

The aim of this paper is to characterize K-contact and Sasakian manifolds whose metrics are generalized quasi-Einstein metric. It is proven that if the metric of a K-contact manifold is generalized quasi-Einstein metric, then the manifold is of constant scalar curvature and in the case of a Sasakian manifold the metric becomes Einstein under certain restriction on the potential function. Several corollaries have been provided. Finally, we consider Sasakian 3-manifold whose metric is generalized quasi-Einstein metric.

• Honam Mathematical Journal
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• v.34 no.2
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• pp.231-239
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• 2012

In this paper, we obtain the Ricci curvature and the scalar curvature on SU(3) with some left invariant Riemannian metric. And then we get a necessary and sufficient condition for the scalar curvature (resp. the Ricci curvature) on the Riemannian manifold SU(3) to be positive.

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