• Communications of the Korean Mathematical Society
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• v.37 no.2
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• pp.595-606
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• 2022

In this paper, we have studied f-biharmonic submanifolds in locally conformal almost cosymplectic space forms and have derived condition on second fundamental form for f-biharmonic submanifolds. Also, we have discussed its integral submanifolds in locally conformal almost cosymplectic space forms.

• Kyungpook Mathematical Journal
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• v.61 no.2
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• pp.395-408
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• 2021

The object of the present paper is to study f-biharmonic submanifolds of three dimensional trans-Sasakian manifolds. We find some necessary and sufficient conditions for such submanifolds to be f-biharmonic.

• Bulletin of the Korean Mathematical Society
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• v.60 no.3
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• pp.705-715
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• 2023

In this paper, we present some new properties for p-biharmonic hypersurfaces in a Riemannian manifold. We also characterize the p-biharmonic submanifolds in an Einstein space. We construct a new example of proper p-biharmonic hypersurfaces. We present some open problems.

• Bulletin of the Korean Mathematical Society
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• v.53 no.3
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• pp.723-732
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• 2016

We characterize proper biharmonic anti-invariant surfaces in 3-dimensional generalized (${\kappa}$, ${\mu}$)-manifolds with constant mean curvature by means of the scalar curvature of the ambient space and the mean curvature. In addition, we give a method for constructing infinity many examples of proper biharmonic submanifolds in a certain 3-dimensional generalized (${\kappa}$, ${\mu}$)-manifold. Moreover, we determine 3-dimensional generalized (${\kappa}$, ${\mu}$)-manifolds which admit a certain kind of proper biharmonic foliation.

• Kyungpook Mathematical Journal
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• v.55 no.1
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• pp.157-168
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• 2015

In this paper, we prove that every f-biharmonic map from a complete Riemannian manifold into a Riemannian manifold with non-positive sectional curvature,satisfying some condition, is f-harmonic. Also we present some properties for the f-biharmonicity of submanifolds of $\mathbb{S}^n$, and we give the classification of f-biharmonic curves in 3-dimensional sphere.

• Kyungpook Mathematical Journal
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• v.59 no.4
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• pp.771-781
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• 2019

In this paper, we consider biharmonic submanifolds of a quaternionic space form. We give the necessary and sufficient conditions for a submanifold to be biharmonic in a quaternionic space form, we study different particular cases for which we obtain some non-existence results and curvature estimates.

• Bulletin of the Korean Mathematical Society
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• v.53 no.6
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• pp.1651-1670
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• 2016

In this paper, we investigate exponentially biharmonic maps u : (M, g) ${\rightarrow}$ (N, h) from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain that if $\int_{M}e^{\frac{p{\mid}r(u){\mid}^2}{2}{\mid}{\tau}(u){\mid}^pdv_g$ < ${\infty}$ ($p{\geq}2$), $\int_{M}{\mid}{\tau}(u){\mid}^2dv_g$ < ${\infty}$ and $\int_{M}{\mid}d(u){\mid}^2dv_g$ < ${\infty}$, then u is harmonic. When u is an isometric immersion, we get that if $\int_{M}e^{\frac{pm^2{\mid}H{\mid}^2}{2}}{\mid}H{\mid}^qdv_g$ < ${\infty}$ for 2 ${\leq}$ p < ${\infty}$ and 0 < q ${\leq}$ p < ${\infty}$, then u is minimal. We also obtain that any weakly convex exponentially biharmonic hypersurface in space form N(c) with $c{\leq}0$ is minimal. These results give affirmative partial answer to conjecture 3 (generalized Chen's conjecture for exponentially biharmonic submanifolds).

• Bulletin of the Korean Mathematical Society
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• v.54 no.6
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• pp.2091-2106
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• 2017

The authors investigate f-biharmonic maps u : (M, g) ${\rightarrow}$ (N, h) from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature, and derive that if $\int_{M}f^p{\mid}{\tau}(u){\mid}^pdv_g$ < ${\infty}$, $\int_{M}{\mid}{\tau}(u){\mid}^2dv_g$ < ${\infty}$ and $\int_{M}{\mid}du{\mid}^2dv_g$ < ${\infty}$, then u is harmonic. When u is an isometric immersion, the authors also get that if u satisfies some integral conditions, then it is minimal. These results give an affirmative partial answer to conjecture 4 (generalized Chen's conjecture for f-biharmonic submanifolds).

• Journal of the Korean Mathematical Society
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• v.52 no.5
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• pp.1097-1108
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• 2015

In this paper, we investigate p-biharmonic maps u : (M, g) $\rightarrow$ (N, h) from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain that if ${\int}_M|{\tau}(u)|^{{\alpha}+p}dv_g$ < ${\infty}$ and ${\int}_M|d(u)|^2dv_g$ < ${\infty}$, then u is harmonic, where ${\alpha}{\geq}0$ is a nonnegative constant and $p{\geq}2$. We also obtain that any weakly convex p-biharmonic hypersurfaces in space formN(c) with $c{\leq}0$ is minimal. These results give affirmative partial answer to Conjecture 2 (generalized Chen's conjecture for p-biharmonic submanifolds).

• Kyungpook Mathematical Journal
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• v.56 no.1
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• pp.273-293
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• 2016

In this paper, we obtain that every biharmonic non-degenerate hypersurfaces in semi-Euclidean space $E^5_s$ with constant scalar curvature of diagonal shape operator has zero mean curvature.