• Journal of the Korean Mathematical Society
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• v.55 no.5
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• pp.1235-1255
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• 2018

We provide definitions for the helicoidal Killing field and the helicoid in arbitrary three-manifolds, and investigate helicoids and ruled minimal surfaces in homogeneous three-manifolds, mainly in $SL_2{\mathbb{R}}$ and Sol(3). In so doing we finish our classification of ruled minimal surfaces in homogeneous three-manifolds with the isometry group of dimension 4.

• Journal of the Korean Mathematical Society
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• v.57 no.4
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• pp.873-892
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• 2020

Thurston, in 1986, discovered that the Schwarzian derivative has mysterious properties similar to the curvature on a manifold. After his work, there are several approaches to develop this notion on Riemannian manifolds. Here, a tensor field is identified in the study of global conformal diffeomorphisms on Finsler manifolds as a natural generalization of the Schwarzian derivative. Then, a natural definition of a Mobius mapping on Finsler manifolds is given and its properties are studied. In particular, it is shown that Mobius mappings are mappings that preserve circles and vice versa. Therefore, if a forward geodesically complete Finsler manifold admits a Mobius mapping, then the indicatrix is conformally diffeomorphic to the Euclidean sphere S^n-1 in ℝⁿ. In addition, if a forward geodesically complete absolutely homogeneous Finsler manifold of scalar flag curvature admits a non-trivial change of Mobius mapping, then it is a Riemannian manifold of constant sectional curvature.

• Journal of applied mathematics & informatics
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• v.40 no.3_4
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• pp.475-490
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• 2022

The main purpose of the present paper is to introduce a brief analysis on some properties of quasi hemi-slant submanifolds of Kenmotsu manifolds. After discussing the introduction and some preliminaries about the Kenmotsu manifold, we worked out some important results in the direction of integrability of the distributions of quasi hemi-slant submanifolds of Kenmotsu manifolds. Afterward, we investigate the conditions for quasi hemi-slant submanifolds of a Kenmotsu manifold to be totally geodesic and later we provide some non-trivial examples to validate the existence of such submanifolds.

• Bulletin of the Korean Mathematical Society
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• v.36 no.4
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• pp.683-691
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• 1999

It has long been a question which homology class is represented by an embedded 2-sphere in a smooth 4-manifold. In this article we study the adjunction inequality, one of main results of Seiberg-Witten theory in smooth 4-manifolds, for an embedded 2-sphere. As a result, we give a criterion which homology class cannot be represented by an embedded 2-sphere in some cases.

• Bulletin of the Korean Mathematical Society
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• v.33 no.4
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• pp.619-629
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• 1996

In studying smooth 4-manifolds the Donaldson invariant has played a central role. In [D1] Donaldson showed that non-vanishing Donaldson invariant of a smooth closed oriented 4-manifold X gives rise to the indecomposability of X. For instance, the complex algebraic suface X cannot decompose to a connected sum $X_1 #X_2$ with both $b_2^+(X_i) > 0$.

• The Pure and Applied Mathematics
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• v.22 no.4
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• pp.359-364
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• 2015

We present smooth simply connected closed 4k-dimensional manifolds N := N_k, for each k ∈ {2, 3, ⋯}, with distinct symplectic deformation equivalence classes [[ω_i]], i = 1, 2. To distinguish [[ω_i]]’s, we used the symplectic Z invariant in [4] which depends only on the symplectic deformation equivalence class. We have computed that Z(N, [[ω₁]]) = ∞ and Z(N, [[ω₂]]) < 0.

• Bulletin of the Korean Mathematical Society
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• v.52 no.4
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• pp.1213-1223
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• 2015

We examine a family of isolated complex surface singularities whose exceptional curves consist of two complex curves with high genera intersecting transversally. Topological data of smoothings of these singularities are determined. We use these computations to construct symplectic 4-manifolds by replacing neighborhoods of the exceptional curves with smoothings of the singularities.

• Journal of the Korean Mathematical Society
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• v.43 no.6
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• pp.1183-1197
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• 2006

Let X be a smooth, closed, connected spin 4-manifold with $b_1(X)=0$ and signature ${\sigma}-(X)$. In this paper we use Seiberg-Witten theory to prove that if X admits a spin alternating $A_4$ action, then $b^+_2(X)$ ${\geq}$ |${\sigma}{(X)}$|/8+3 under some non-degeneracy conditions.

• Bulletin of the Korean Mathematical Society
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• v.35 no.1
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• pp.1-12
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• 1998

We construct exotic symplectic structures on $S^3 \times R$ which is obtained by the symplectic sum of two smooth symplectic four-manifolds with exotic symplectic structures, each of which is diffeomorphic to $R^4$.

• Journal of the Korean Mathematical Society
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• v.53 no.4
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• pp.769-780
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• 2016

In this paper, we first prove some Liouville type theorems for elliptic inequalities on weighted manifolds which support a weighted Sobolev-type inequality. Secondly, applying the Liouville type theorems to self-shrinkers, we obtain some global rigidity theorems.