• Journal of the Korean Mathematical Society
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• v.37 no.4
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• pp.503-519
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• 2000

It is well-known that an analytic generic CR submainfold M of codimension m in Cn+m is locally transformed by a biholomorphic mapping to a plane Cn$\times$Rm ⊂ Cn$\times$Cm whenever the Levi form L on M vanishes identically. We obtain such a normalizing biholomorphic mapping of M in terms of the defining function of M. Then it is verified without Frobenius theorem that M is locally foliated into complex manifolds of dimension n.

• Bulletin of the Korean Mathematical Society
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• v.56 no.1
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• pp.169-178
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• 2019

Let $M^{2n+1}$, $n{\geq}1$, be a smooth manifold with a pseudoconvex integrable CR structure of hypersurface type. We consider a sequence of CR invariant subsets $M={\mathcal{S}}_0{\supset}{\mathcal{S}}_1{\supset}{\cdots}{\supset}{\mathcal{S}}_n$, where $S_q$ is the set of points where the Levi-form has nullity ${\geq}q$. We prove that ${\mathcal{S}}{_q}^{\prime}s$ are locally given as common zero sets of the coefficients $A_j$, $j=0,1,{\ldots},q-1$, of the characteristic polynomial of the Levi-form. Some sufficient conditions for local existence of complex submanifolds are presented in terms of the coefficients $A_j$.

• Journal of the Korean Mathematical Society
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• v.39 no.3
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• pp.425-437
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• 2002

Let $\Omega$ be a smoothly bounded pseudoconvex domain in $C^{n}$ and let $z^{0}$ $\in$b$\Omega$ a point of finite type. We also assume that the Levi form of b$\Omega$ is comparable in a neighborhood of $z^{0}$ . Then we get precise estimates of the Bergman kernel function, $K_{\Omega}$(z, w), and its derivatives in a neighborhood of $z^{0}$ . .

• Bulletin of the Korean Mathematical Society
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• v.58 no.3
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• pp.781-794
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• 2021

Let Ω be a smoothly bounded pseudoconvex domain in ℂⁿ and assume that the (n - 2)-eigenvalues of the Levi-form are comparable in a neighborhood of z₀ ∈ bΩ. Also, assume that there is a smooth 1-dimensional analytic variety V whose order of contact with bΩ at z₀ is equal to 𝜂 and 𝚫_n-2(z₀) < ∞. We show that the maximal gain in Hölder regularity for solutions of the ${\bar{\partial}}$-equation is at most ${\frac{1}{\eta}}$.

• Journal of the Korean Mathematical Society
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• v.47 no.5
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• pp.1001-1015
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• 2010

Let M be a $C^{\infty}$ real hypersurface in $\mathbb{C}^{n+1}$, $n\;{\geq}\;1$, locally given as the zero locus of a $C^{\infty}$ real valued function r that is defined on a neighborhood of the reference point $P\;{\in}\;M$. For each k = 1,..., n we present a necessary and sufficient condition for there to exist a complex manifold of dimension k through P that is contained in M, assuming the Levi form has rank n - k at P. The problem is to find an integral manifold of the real 1-form $i{\partial}r$ on M whose tangent bundle is invariant under the complex structure tensor J. We present generalized versions of the Frobenius theorem and make use of them to prove the existence of complex submanifolds.

• Journal of the Korean Mathematical Society
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• v.39 no.1
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• pp.91-102
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• 2002

Let M and N be CR manifolds with nondegenerate Levi forms of hypersurface type of dimension 2m ＋ 1 and 2n ＋ 1, respectively, where 1 $\leq$ m $\leq$ n. Let f : M longrightarrow N be a CR mapping. Under a generic assumption we construct a complete system of finite order for the infinitesimal deformations of f. In particular, we prove the space of infinitesimal deformations of f forms a finite dimensional Lie algebra.

• East Asian mathematical journal
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• v.26 no.1
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• pp.75-79
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• 2010

Let G be a compact connected semisimple Lie group, g the Lie algebra of G, g the canonical metric (the biinvariant Riemannian metric which is induced from the Killing form of g), and $\nabla$ be the Levi-Civita connection for the metric g. Then, we get the fact that the Levi-Civita connection $\nabla$ in the tangent bundle TG over (G, g) is a Yang-Mills connection.

• Bulletin of the Korean Mathematical Society
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• v.54 no.2
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• pp.543-557
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• 2017

In this paper, we investigate the notion of q-pseudoconvexity to discuss and describe some geometric characterizations of q-pseudoconvex domains ${\Omega}{\subset}{\mathbb{C}}^n$. In particular, we establish that ${\Omega}$ is q-pseudoconvex, if and only if, for every boundary point, the Levi form of the boundary is semipositive on the intersection of the holomorphic tangent space to the boundary with any (n-q+1)-dimensional subspace $E{\subset}{\mathbb{C}}^n$. Furthermore, we prove that the Kiselman's minimum principal holds true for all q-pseudoconvex domains in ${\mathbb{C}}^p{\times}{\mathbb{C}}^n$ such that each slice is a convex tube in ${\mathbb{C}}^n$.

• Journal of the Chungcheong Mathematical Society
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• v.23 no.4
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• pp.813-821
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• 2010

Let G and H be compact connected Lie groups with biinvariant Riemannian metrics g and h respectively, ${\phi}$ a group isomorphism of G onto H, and $E:={\phi}^{-1}TH$ the induced bundle by $\phi$ over the base manifold G of the tangent bundle TH of H. Let ${\nabla}$ and $^H{\nabla}$ be the Levi-Civita connections for the metrics g and h respectively, $\tilde{\nabla}$ the induced connection by the map ${\phi}$ and $^H{\nabla}$. Then, a necessary and sufficient condition for $\tilde{\nabla}$ in the bundle (${\phi}^{-1}TH$, G, ${\pi}$) to be a Yang- Mills connection is the fact that the Levi-Civita connection ${\nabla}$ in the tangent bundle over (G, g) is a Yang- Mills connection. As an application, we get the following: Let ${\psi}$ be an automorphism of a compact connected semisimple Lie group G with the canonical metric g (the metric which is induced by the Killing form of the Lie algebra of G), ${\nabla}$ the Levi-Civita connection for g. Then, the induced connection $\tilde{\nabla}$, by ${\psi}$ and ${\nabla}$, is a Yang-Mills connection in the bundle (${\phi}^{-1}TH$, G, ${\pi}$) over the base manifold (G, g).

• Journal of the Korean Mathematical Society
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• v.40 no.4
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• pp.727-745
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• 2003

We construct strongly pseudoconvex handlebodies in $C^{n}$ whose center is a quadratic strongly pseudoconvex domain with an attached flat Lagrangian disc or plane.