• Journal of the Korean Mathematical Society
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• v.55 no.3
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• pp.735-747
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• 2018

Let M be a real hypersurface of a complex space form with constant curvature c. In this paper, we study the hypersurface M admitting Miao-Tam critical metric, i.e., the induced metric g on M satisfies the equation: $-({\Delta}_g{\lambda})g+{\nabla}^2_g{\lambda}-{\lambda}Ric=g$, where ${\lambda}$ is a smooth function on M. At first, for the case where M is Hopf, c = 0 and $c{\neq}0$ are considered respectively. For the non-Hopf case, we prove that the ruled real hypersurfaces of non-flat complex space forms do not admit Miao-Tam critical metrics. Finally, it is proved that a compact hypersurface of a complex Euclidean space admitting Miao-Tam critical metric with ${\lambda}$ > 0 or ${\lambda}$ < 0 is a sphere and a compact hypersurface of a non-flat complex space form does not exist such a critical metric.

• Bulletin of the Korean Mathematical Society
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• v.60 no.4
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• pp.1085-1100
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• 2023

For any given function f, we focus on the so-called prescribed mean curvature problem for the measure e^-f(|x|2)dx provided thate^-f(|x|2) ∈ L¹(ℝⁿ⁺¹). More precisely, we prove that there exists a smooth hypersurface M whose metric is ds² = dρ² + ρ²d𝜉² and whose mean curvature function is ${\frac{1}{n}}(\frac{u^p}{{\rho}^{\beta}})e^{f({\rho}^2)}{\psi}(\xi)$ for any given real constants p, β and functions f and ψ where u and ρ are the support function and radial function of M, respectively. Equivalently, we get the existence of a smooth solution to the following quasilinear equation on the unit sphere 𝕊ⁿ, $${\sum_{i,j}}({{\delta}_{ij}-{\frac{{\rho}_i{\rho}_j}{{\rho}^2+|{\nabla}{\rho}|^2}})(-{\rho}ji+{\frac{2}{{\rho}}}{\rho}j{\rho}i+{\rho}{\delta}_{ji})={\psi}{\frac{{\rho}^{2p+2-n-{\beta}}e^{f({\rho}^2)}}{({\rho}^2+|{\nabla}{\rho}|^2)^{\frac{p}{2}}}}$$ under some conditions. Our proof is based on the powerful method of continuity. In particular, if we take $f(t)={\frac{t}{2}}$, this may be prescribed mean curvature problem in Gauss measure space and it can be seen as an embedded result in Gauss measure space which will be needed in our forthcoming papers on the differential geometric analysis in Gauss measure space, such as Gauss-Bonnet-Chern theorem and its application on positive mass theorem and the Steiner-Weyl type formula, the Plateau problem and so on.

• East Asian mathematical journal
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• v.30 no.1
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• pp.69-77
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• 2014

In this paper, we introduce a new generalized resolvent in a Banach space and discuss its some properties. Using these properties, we obtain an iterative scheme for finding a point which is a fixed point of relatively weak nonexpansive mapping and a zero of monotone mapping. Furthermore, strong convergence of the scheme to a point which is a fixed point of relatively weak nonexpansive mapping and a zero of monotone mapping is proved.

• Journal of the Korean Mathematical Society
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• v.37 no.6
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• pp.1007-1019
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• 2000

In this paper, we discuss a uniqueness problem for the Cauchy problem of the Euler equation. W give a sufficient condition on the vorticity to show the uniqueness of a class of generalized solution in terms of the generalized solution in terms o the generalized Besov space. The condition allows the iterated logarithmic singularity to the vorticity of the solution. We also discuss the break down (or blow up) condition for a smooth solution to the Euler equation under the related assumption.

• Journal of Astronomy and Space Sciences
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• v.31 no.3
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• pp.199-204
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• 2014

This paper intends to highlight the effect of the third-body in an inclined orbit on a spacecraft orbiting the primary mass. To achieve this goal, a new origin of coordinate is introduced in the primary and the X-axis toward the node of the spacecraft. The disturbing function is expanded up to the second order using Legendre polynomials. A double-averaged analytical model is exploited to produce the evolutions of mean orbital elements as smooth curves.

• Journal of the Korean Mathematical Society
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• v.40 no.3
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• pp.503-516
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• 2003

We show in this paper that every domain in a separable Hilbert space, say H, which has a $C^2$ smooth strongly pseudoconvex boundary point at which an automorphism orbit accumulates is biholomorphic to the unit ball of H. This is the complete generalization of the Wong-Rosay theorem to a separable Hilbert space of infinite dimension. Our work here is an improvement from the preceding work of Kim/Krantz [10] and subsequent improvement of Byun/Gaussier/Kim [3] in the infinite dimensions.

• Communications of the Korean Mathematical Society
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• v.13 no.4
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• pp.825-838
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• 1998

We prove that if a real hypersurface with constant mean curvature of a complex space form satisfying ▽$_{ξ/}$S = 0 and Sξ = $\sigma$ξ for a smooth function $\sigma$, then the structure vector field ξ is principal, where S denotes the Ricci tensor of the hypersurface.

• Journal of the Korean Mathematical Society
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• v.43 no.6
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• pp.1143-1158
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• 2006

We consider the projectivization of Minkowski space with the analytic continuation of the hyperbolic metric and call this an extended hyperbolic space. We can measure the volume of a domain lying across the boundary of the hyperbolic space using an analytic continuation argument. In this paper we show this method can be further generalized to find the volume of a domain with smooth boundary with suitable regularity in dimension 2 and 3. We also discuss that this volume is invariant under the group of hyperbolic isometries and that this regularity condition is sharp.

• Journal of information and communication convergence engineering
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• v.8 no.4
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• pp.411-415
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• 2010

Conventional Augmented Reality has used data gloves or markers for smooth interaction between objects and background. This causes inconvenience of use and lower immersion. To build up immersion in Augmented Reality, additional input devices must be removed. For this purpose, accurate recognition of space coordinates is needed even with no attachment of markers. This paper proposes a method to create virtual space coordinates for interaction without wearing additional input devices so as to improve immersion in Augmented Reality. The acquired image was projected to 2D space and vanishing lines were extracted to calculate the virtual space coordinates. Then the sizes of the inserted objects were varied in accordance with the size of the virtual coordinates area based on the image projected onto the 2D coordinates. This resulted in improved immersion. This method can increase the efficiency of object creation by excluding the use of a 3D modeler for creation of 3D objects.

• Journal of the Korean Association of Oral and Maxillofacial Surgeons
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• v.31 no.1
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• pp.46-54
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• 2005

Purpose: This study was performed in order to compare the osseointegration of 4 different surfaced implants in the dog's tibia which has thick dense cortical bone and loose marrow space. Materials & methods: Four mongrel dogs and four different surface types of implants, smooth surfaced AVANA implants, RBM surfaced AVANA implants, HA-coated Steri-Oss implants and SLA Bicon implants, were used in this study. The animals were divided into 4 groups on the basis of implant surface characteristics: Control group, RBM group, HA group, and SLA group. Three implants of each group were installed into the metaphysis of tibia of adult dogs. The animals were sacrificed at 8 weeks after implantation. The undecalcified specimens were prepared for histological examination and histomorphometric analysis of implant-bone contact ratios. Results: Radiographically and histologically good osseointegration of implant was observed in the dense cortical bone, but poor osseointegration was observed in the marrow space. Histologically more bone apposition to implant surface was found in rough surfaced groups than the smooth surfaced, Control group. In histomorphometric findings of cortical bone the average bone-implant contact ratios of HA group (95.4%, p<0.01), RBM group (87.1%, p<0.05), and SLA group (86.0%, p<0.05) were significantly higher than that of Control group (75.9%). In marrow space the average bone-implant contact ratios of HA group (76.1%, p<0.01) and SLA group (45.4%, p<0.05) were significantly higher than that of Control group (29.6%). The ratio of RBM group was higher than that of Control group but there was no significantly difference between RBM group and Control group. Conclusion: These results suggest that the rough surfaced implants can obtain the better osseointegration than the smooth surfaced implant in the cortical and marrow space and that HA-coated implants can obtain the best osseointegration in the marrow space among them.