• Bulletin of the Korean Mathematical Society
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• v.53 no.4
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• pp.1113-1122
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• 2016

Let (M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Yamabe flow. In the paper we derive the evolution for the first eigenvalue of geometric operator $-{\Delta}_{\phi}+{\frac{R}{2}}$ under the Yamabe flow, where ${\Delta}_{\phi}$ is the Witten-Laplacian operator, ${\phi}{\in}C^2(M)$, and R is the scalar curvature with respect to the metric g(t). As a consequence, we construct some monotonic quantities under the Yamabe flow.

• Kyungpook Mathematical Journal
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• v.59 no.2
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• pp.341-352
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• 2019

Let M be an n-dimensional closed Riemannian manifold with metric g, $d{\mu}=e^{-{\phi}(x)}d{\nu}$ be the weighted measure and ${\Delta}_{p,{\phi}}$ be the weighted p-Laplacian. In this article we will study the evolution and monotonicity for the first nonzero eigenvalue problem of the weighted p-Laplace operator acting on the space of functions along the Yamabe flow on closed Riemannian manifolds. We find the first variation formula of it along the Yamabe flow. We obtain various monotonic quantities and give an example.

• Bulletin of the Korean Mathematical Society
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• v.55 no.3
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• pp.851-863
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• 2018

We derive lower bounds of the scalar curvature on complete non-compact gradient Yamabe solitons under some integral curvature conditions. Based on this, we prove that potential functions of Yamabe solitons have at most quadratic growth for distance function. We also obtain a finite topological type property on complete shrinking gradient Yamabe solitons under suitable scalar curvature assumptions.

• Honam Mathematical Journal
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• v.43 no.4
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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.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.23 no.4
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• pp.351-365
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• 2019

This article presents a new approach for conformal flattening with optimal cone singularity. The algorithm here takes an optimal control for selecting optimal cones and uses the Ricci flow to force the flattening. This work is considered as a modification to the work of Soliman et al. [1] in the sense that they make use of the Yamabe equation for the flattening, which is an approximation of the Ricci flow. We present a numerical algorithm based on the optimal control with the mathematical background. Several numerical results validate that our method is optimal in total cone angle and usage of the Ricci flow ensures the conformal flattening while selecting optimal cones.