• 대한수학회지
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• 제54권4호
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• pp.1189-1208
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• 2017

In this paper, we study vanishing properties for $L^2$ harmonic 1-forms on a gradient shrinking Ricci soliton. We prove that if (M, g, f) is a complete oriented noncompact gradient shrinking Ricci soliton with potential function f, then there are no non-trivial $L^2$ harmonic 1-forms which are orthogonal to df. Second, we show that if the scalar curvature of the metric g is greater than or equal to (n - 2)/2, then there are no non-trivial $L^2$ harmonic 1-forms on (M, g). We also show that any multiplication of the total differential df by a function cannot be an $L^2$ harmonic 1-form unless it is trivial. Finally, we derive various integral properties involving the potential function f and $L^2$ harmonic 1-forms, and handle their applications.

• 대한수학회지
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• 제55권1호
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• pp.161-174
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• 2018

The aim of this article is to study the k-almost Ricci soliton and k-almost gradient Ricci soliton on contact metric manifold. First, we prove that if a compact K-contact metric is a k-almost gradient Ricci soliton, then it is isometric to a unit sphere $S^{2n+1}$. Next, we extend this result on a compact k-almost Ricci soliton when the flow vector field X is contact. Finally, we study some special types of k-almost Ricci solitons where the potential vector field X is point wise collinear with the Reeb vector field ${\xi}$ of the contact metric structure.

• 대한수학회논문집
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• 제35권2호
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• pp.613-623
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• 2020

The notion of quasi-Einstein metric in theoretical physics and in relation with string theory is equivalent to the notion of Ricci soliton in differential geometry. Quasi-Einstein metrics or Ricci solitons serve also as solution to Ricci flow equation, which is an evolution equation for Riemannian metrics on a Riemannian manifold. Quasi-Einstein metrics are subject of great interest in both mathematics and theoretical physics. In this paper the notion of Ricci 𝜌-soliton as a generalization of Ricci soliton is defined. We are motivated by the Ricci-Bourguignon flow to define this concept. We show that if a 3-dimensional almost Kenmotsu Einstein manifold M is a 𝜌-soliton, then M is a Kenmotsu manifold of constant sectional curvature -1 and the 𝜌-soliton is expanding with λ = 2.

• Kyungpook Mathematical Journal
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• 제58권2호
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• pp.389-398
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• 2018

In this paper we obtain the condition for the existence of Ricci solitons in nonflat quaternion space form by using Eisenhart problem. Also it is proved that if (g, V, ${\lambda}$) is Ricci soliton then V is solenoidal if and only if it is shrinking, steady and expanding depending upon the sign of scalar curvature. Further it is shown that Ricci soliton in semi-symmetric quaternion space form depends on quaternion sectional curvature c if V is solenoidal.

• Journal of the Optical Society of Korea
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• 제16권4호
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• pp.425-431
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• 2012

Utilizing the three-dimensional Snyder-Mitchell model with a PT-symmetric potential, we study the influence of PT symmetry on beam propagation in strongly nonlocal nonlinear media. The complex Coulomb potential is used as the PT-symmetric potential. A localized spatiotemporal accessible soliton solution of the model is obtained. Specific values of the modulation depth for different soliton parameters are discussed. Our results reveal that in these media the localized solitons can exist in various shapes, such as single-layer and multi-layer disk-shaped structures, as well as vortex-ring and necklace patterns.

• Kyungpook Mathematical Journal
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• 제61권3호
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• pp.645-660
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• 2021

This paper examines the behavior of a 3-dimensional trans-Sasakian manifold equipped with a gradient generalized quasi-Yamabe soliton. In particular, It is shown that α-Sasakian, β-Kenmotsu and cosymplectic manifolds satisfy the gradient generalized quasi-Yamabe soliton equation. Furthermore, in the particular case when the potential vector field ζ of the quasi-Yamabe soliton is of gradient type ζ = grad(ψ), we derive a Poisson's equation from the quasi-Yamabe soliton equation. Also, we study harmonic aspects of quasi-Yamabe solitons on 3-dimensional trans-Sasakian manifolds sharing a harmonic potential function ψ. Finally, we observe that 3-dimensional compact trans-Sasakian manifold admits the gradient generalized almost quasi-Yamabe soliton with Hodge-de Rham potential ψ. This research ends with few examples of quasi-Yamabe solitons on 3-dimensional trans-Sasakian manifolds.

• 대한수학회논문집
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• 제34권1호
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• pp.321-331
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• 2019

The present paper deals with a study of infinitesimal CL-transformations on Kenmotsu manifolds, whose metric is Yamabe soliton and obtained sufficient conditions for such solitons to be expanding, steady and shrinking. Among others, we find a necessary and sufficient condition of a Yamabe soliton on Kenmotsu manifold with respect to CL-connection to be Yamabe soliton on Kenmotsu manifold with respect to Levi-Civita connection. We found the necessary and sufficient condition for the Yamabe soliton structure to be invariant under Schouten-Van Kampen connection. Finally, we constructed an example of steady Yamabe soliton on 3-dimensional Kenmotsu manifolds with respect to Schouten-Van Kampen connection.

• 대한수학회논문집
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• 제37권2호
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• pp.585-594
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• 2022

In this article we classify four dimensional gradient Ricci solitons (M, g, f) with half harmonic Weyl curvature and at most two distinct Ricci-eigenvalues at each point. Indeed, we showed that, in a neighborhood V of each point in some open dense subset of M, (V, g) is isometric to one of the following: (i) an Einstein manifold. (ii) a domain in the Riemannian product (ℝ², g₀) × (N, ${\tilde{g}}$), where g₀ is the flat metric on ℝ² and (N, ${\tilde{g}}$) is a two dimensional Riemannian manifold of constant curvature λ ≠ 0. (iii) a domain in ℝ × W with the warped product metric $ds^2+h(s)^2{\tilde{g}}$, where ${\tilde{g}}$ is a constant curved metric on a three dimensional manifold W.

• 호남수학학술지
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• 제43권4호
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• pp.613-626
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• 2021

The aim of the present work is to study the properties of three-dimensional Lorentzian para-Kenmotsu manifolds equipped with a Yamabe soliton. It is proved that every three-dimensional Lorentzian para-Kenmotsu manifold is Ricci semi-symmetric if and only if it is Einstein. Also, if the metric of a three-dimensional semi-symmetric Lorentzian para-Kenmotsu manifold is a Yamabe soliton, then the soliton is shrinking and the flow vector field is Killing. We also study the properties of three-dimensional Ricci symmetric and 𝜂-parallel Lorentzian para-Kenmotsu manifolds with Yamabe solitons. Finally, we give a non-trivial example of three-dimensional Lorentzian para-Kenmotsu manifold.

• 대한수학회보
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• 제60권1호
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• pp.171-184
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• 2023

In this paper, we show that a complete translating soliton Σ^m in ℝⁿ for the mean curvature flow is stable with respect to weighted volume functional if Σ satisfies that the L^m norm of the second fundamental form is smaller than an explicit constant that depends only on the dimension of Σ and the Sobolev constant provided in Michael and Simon [12]. Under the same assumption, we also prove that under this upper bound, there is no non-trivial f-harmonic 1-form of L²_f on Σ. With the additional assumption that Σ is contained in an upper half-space with respect to the translating direction then it has only one end.