• 대한수학회논문집
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• 제30권4호
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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.

• 대한수학회지
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• 제49권5호
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• pp.965-976
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• 2012

We prove an $L^2$-estimate of Ricci curvature in terms of harmonic 1-forms on a closed oriented Riemannian 3-manifold admitting a solution of any rescaled Seiberg-Witten equations. We also give a necessary condition to be a monopole class on some special connected sums.

• 대한수학회보
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• 제39권2호
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• pp.327-332
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• 2002

In this paper we prove that if M is a J-invariant sub-manifold of codimension 2 in a complex projective space $P_{n+1}(C)$, and the second fundamental tensor is cyclic-parallel or M has harmonic curvature, then M is locally complex quadric Q$_n$(C) or P$_n$(C).

• 대한수학회보
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• 제46권6호
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• pp.1213-1219
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• 2009

Let M$^n$ be a complete oriented non-compact minimally immersed submanifold in a complete Riemannian manifold N$^{n+p}$ of nonnegative curvature. We prove that if M is super-stable, then there are no non-trivial L$^2$ harmonic one forms on M. This is a generalization of the main result in [8].

• Journal of applied mathematics & informatics
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• 제7권3호
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• pp.1039-1044
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• 2000

We investigate the projectively flat warped product manifolds and study the geometric structure of the base space and its fibre. Specifically we find the conditions that the scalar curvature of the base space (B,g) vanishes if and only if f is harmonic on (B,g) and the fibre (F,$\bar{g}$) is a space of constant curvature.

• 대한수학회지
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• 제57권2호
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• pp.539-552
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• 2020

In this paper we consider gradient almost Ricci solitons with weak conditions on Weyl and Bach tensors. We show that a gradient almost Ricci soliton has harmonic Weyl curvature if it has fourth order divergence-free Weyl tensor, or it has divergence-free Bach tensor. Furthermore, if its Weyl tensor is radially flat, we prove such a gradient almost Ricci soliton is locally a warped product with Einstein fibers. Finally, we prove a rigidity result on compact gradient almost Ricci solitons satisfying an integral condition.

• 대한수학회보
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• 제34권1호
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• pp.103-114
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• 1997

The asymptotic Dirichlet problem for harmonic functions on a noncompact complete Riemannian manifold has a long history. It is to find the harmonic function satisfying the given Dirichlet boundary condition at infinity. By now, it is well understood [A, AS, Ch, S], when M is a Cartan-Hadamard manifold with sectional curvature $-b^2 \leq K_M \leq -a^2 < 0$. (By a Cartan-Hadamard manifold, we mean a complete simply connected manifold of non-positive sectional curvature.)

• 대한수학회보
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• 제59권5호
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• pp.1167-1176
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• 2022

In this paper, we study Einstein-type manifolds generalizing static spaces and V-static spaces. We prove that if an Einstein-type manifold has non-positive complete divergence of its Weyl tensor and non-negative complete divergence of Bach tensor, then M has harmonic Weyl curvature. Also similar results on an Einstein-type manifold with complete divergence of Riemann tensor are proved.

• Korean Journal of Mathematics
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• 제31권2호
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• pp.139-144
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• 2023

In this paper, we show that a recurrent manifold with harmonic curvature tensor is locally symmetric and that an Einstein and conformally recurrent manifold is locally symmetric. As a consequence, Einstein and recurrent manifolds must be locally symmetric. On the other hand, we have obtained some results for a (conformally) recurrent manifold with parallel vector field and also investigated some results for a (conformally) recurrent manifold with concircular vector field.

• 대한수학회지
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• 제55권3호
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• pp.553-574
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• 2018

In this paper, we study harmonic maps and biharmonic maps on the principal G-bundle in Kobayashi and Nomizu and also the warped product $P=M{\times}_fF$ for a $C^{\infty}$(M) function f on M studied by Bishop and O'Neill, and Ejiri.