• Bulletin of the Korean Mathematical Society
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• v.56 no.5
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• pp.1327-1340
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• 2019

We establish a monotonicity formula of V-harmonic maps by using the stress-energy tensor. Use the monotonicity formula, we can derive a Liouville type theorem for V-harmonic maps. As applications, we also obtain monotonicity and constancy of Weyl harmonic maps from conformal manifolds to Riemannian manifolds and ${\pm}holomorphic$ maps between almost Hermitian manifolds. Finally, a constant boundary-value problem of V-harmonic maps is considered.

• Honam Mathematical Journal
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• v.41 no.2
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• pp.343-356
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• 2019

In this paper, we introduce metallic maps between metallic Riemannian manifolds, provide an example and obtain certain conditions for such maps to be totally geodesic. We also give a sufficient condition for a map between metallic Riemannian manifolds to be harmonic map. Then we investigate the constancy of certain maps between metallic Riemannian manifolds and various manifolds by imposing the holomorphic-like condition. Moreover, we check the reverse case and show that some such maps are constant if there is a condition for this.

• Journal of the Korean Mathematical Society
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• v.44 no.5
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• pp.1079-1092
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• 2007

This paper studies the deformation theory of a holomorphic surjective map from a normal compact complex space X to a compact $K\"{a}hler$ manifold Y. We will show that when the target has non-negative Kodaira dimension, all deformations of surjective holomorphic maps $X{\rightarrow}Y$ come from automorphisms of an unramified covering of Y and the underlying reduced varieties of associated components of Hol(X, Y) are complex tori. Under the additional assumption that Y is projective algebraic, this was proved in [7]. The proof in [7] uses the algebraicity in an essential way and cannot be generalized directly to the $K\"{a}hler$ setting. A new ingredient here is a careful study of the infinitesimal deformation of orbits of an action of a complex torus. This study, combined with the result for the algebraic case, gives the proof for the $K\"{a}hler$ setting.

• Bulletin of the Korean Mathematical Society
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• v.42 no.1
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• pp.157-164
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• 2005

Let $RRat_k$ ($CP^n$) denote the space of basepoint-preserving conjugation-equivariant holomorphic maps of degree k from $S^2$ to $CP^n$. A map f ; $S^2 {\to}CP^n$ is said to be full if its image does not lie in any proper projective subspace of $CP^n$. Let $RF_k(CP^n)$ denote the subspace of $RRat_k(CP^n)$ consisting offull maps. In this paper we determine $H{\ast}(RF_k(CP^2); Z/p)$ for all primes p.

• Communications of the Korean Mathematical Society
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• v.30 no.4
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• pp.471-479
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• 2015

In this paper, we prove that any stable f-harmonic map ${\psi}$ from ${\mathbb{S}}^2$ to N is a holomorphic or anti-holomorphic map, where N is a $K{\ddot{a}}hlerian$ manifold with non-positive holomorphic bisectional curvature and f is a smooth positive function on the sphere ${\mathbb{S}}^2$with Hess $f{\leq}0$. We also prove that any stable f-harmonic map ${\psi}$ from sphere ${\mathbb{S}}^n$ (n > 2) to Riemannian manifold N is constant.

• Journal of the Korean Mathematical Society
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• v.46 no.5
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• pp.895-905
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• 2009

Let $B^{n+1}$ be the unit ball in the complex vector space $\mathbb{C}^{n+1}$ with the standard Hermitian metric. Let ${\Sigma}^n={\partial}B^{n+1}=S^{2n+1}$ be the boundary sphere with the induced CR structure. Let f : ${\Sigma}^n{\hookrightarrow}{\Sigma}^N$ be a local CR immersion. If N < 3n - 1, the asymptotic vectors of the CR second fundamental form of f at each point form a subspace of the CR(horizontal) tangent space of ${\Sigma}^n$ of codimension at most 1. We study the higher order derivatives of this relation, and we show that a linearly full local CR immersion f : ${\Sigma}^n{\hookrightarrow}{\Sigma}^N$, N $\leq$ 3n-2, can only occur when N = n, 2n, or 2n + 1. As a consequence, it gives an extension of the classification of the rational proper holomorphic maps from $B^{n+1}$ to $B^{2n+2}$ by Hamada to the classification of the rational proper holomorphic maps from $B^{n+1}$ to $B^{3n+1}$.

• The Pure and Applied Mathematics
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• v.21 no.3
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• pp.219-227
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• 2014

In this note we study the Schwarz lemma and inequalities for some holomorphic functions on the unit disc. Also, we obtain the inequality of the derivative of holomorphic maps at a boundary point of the unit disc and find a holomorphic map to satisfy the equality.

• Kyungpook Mathematical Journal
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• v.48 no.1
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• pp.155-163
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• 2008

This is an extended version of the paper [K] of the author. The average formulas on the circles and disks around arbitrary points of Nevanlinna counting functions of holomorphic self-maps of the unit disk, given in terms of the boundary values of the selfmaps, are shown to give another characterization of the whole class or a special subclass of inner functions in terms of Nevanlinna counting function in addition to the previous applications to Rudin's orthogonal functions.

• Bulletin of the Korean Mathematical Society
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• v.59 no.6
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• pp.1423-1438
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• 2022

In this paper, we study a special exhaustion function on almost Hermitian manifolds and establish the existence result by using the Hessian comparison theorem. From the viewpoint of the exhaustion function, we establish a related Schwarz type lemma for almost holomorphic maps between two almost Hermitian manifolds. As corollaries, we deduce several versions of Schwarz and Liouville type theorems for almost holomorphic maps.

• Bulletin of the Korean Mathematical Society
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• v.59 no.2
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• pp.375-395
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• 2022

We define an almost Norden submersion (holomorphic and semi-Riemannian submersion) between almost Norden manifolds and show that, in most of the cases, the base manifold has the similar kind of structure as that of total manifold. We obtain necessary and sufficient conditions for almost Norden submersion to be a totally geodesic map. We also derive decomposition theorems for the total manifold of such submersions. Moreover, we study the harmonicity of almost Norden submersions between almost Norden manifolds and between Kaehler-Norden manifolds. Finally, we derive conditions for an almost Norden submersion to be a harmonic morphism.