• 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.

• Bulletin of the Korean Mathematical Society
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• v.26 no.2
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• pp.175-177
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• 1989

The following is the basic problem about the stability in Riemannian Geometry; given a Riemannian manifold N, find all stable complete minimal submanifolds of N. As answers of this problem, do Carmo-Peng [1] and Fischer-Colbrie and Schoen [3] showed that the stable minimal surfaces in R$^{3}$ are planes and Schoen-Yau [5] and Fischer-Colbrie and Schoen [3] gave a solution for the case where the ambient space is a three dimensional manifold with nonnegative scalar curvature. In this paper we will remove the assumption of finite absolute total curvature in [3, Theorem 3].

• 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.

• Bulletin of the Korean Mathematical Society
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• v.17 no.2
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• pp.109-113
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• 1981

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

• The Journal of Engineering Geology
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• v.25 no.1
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• pp.1-8
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• 2015

Deep geological disposal is the preferred storage method for high-level radioactive waste, because it ensures stable long-term storage with minimal potential for human disruption. Because of the risk of groundwater contamination, a buffer of steel and bentonite layers has been proposed to prevent the leaching of radionuclides into groundwater. Quartz is one of the most common minerals in earth's crust. To understand how deformation and dissolution phenomena affect waste disposal, here we study quartz samples at pressure, temperature, and pH conditions typical of deep geological disposal sites. We perform a dissolution experiment for single quartz crystals under different pressure and temperature conditions. Solution samples are collected and the dissolution rate is calculated by analyzing Si concentrations in a solution excited by inductively coupled plasma-atomic emission spectroscopy (ICP-AES). After completing the dissolution experiment, deformation of the quartz sample surfaces is investigated with a confocal laser scanning microscope (CLSM). An empirical formula is introduced that describes the relationship between dissolution rate, pressure, and temperature. These results suggest that bentonite layers in engineering barrier systems may be vulnerable to thermal deformation, even when exposed to higher temperatures on relatively short timescales.