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EIGENVALUE MONOTONICITY OF (p, q)-LAPLACIAN ALONG THE RICCI-BOURGUIGNON FLOW
Azami, Shahroud Korean Mathematical Society 2019 대한수학회논문집 Vol.34 No.1
In this paper we study monotonicity the first eigenvalue for a class of (p, q)-Laplace operator acting on the space of functions on a closed Riemannian manifold. We find the first variation formula for the first eigenvalue of a class of (p, q)-Laplacians on a closed Riemannian manifold evolving by the Ricci-Bourguignon flow and show that the first eigenvalue on a closed Riemannian manifold along the Ricci-Bourguignon flow is increasing provided some conditions. At the end of paper, we find some applications in 2-dimensional and 3-dimensional manifolds.
Shahroud Azami 호남수학회 2023 호남수학학술지 Vol.45 No.4
In this paper, we study quasi-Sasakian 3-dimensional manifolds admitting generalized η-Ricci solitons associated to the Schoutenvan Kampen connection. We give an example of generalized η-Ricci solitons on a quasi-Sasakian 3-dimensional manifold with respect to the Schouten-van Kampen connection to prove our results.
THE FIRST EIGENVALUE OF SOME (p, q)-LAPLACIAN AND GEOMETRIC ESTIMATES
Azami, Shahroud Korean Mathematical Society 2018 대한수학회논문집 Vol.33 No.1
We study the nonlinear eigenvalue problem for some of the (p, q)-Laplacian on compact manifolds with zero boundary condition. In particular, we obtain some geometric estimates for the first eigenvalue.
Generalized hyperbolic geometric flow
Shahroud Azami,Ghodratallah Fasihi-Ramandi,Vahid Pirhadi 대한수학회 2023 대한수학회논문집 Vol.38 No.2
In the present paper, we consider a kind of generalized hyperbolic geometric flow which has a gradient form. Firstly, we establish the existence and uniqueness for the solution of this flow on an $n$-dimensional closed Riemannian manifold. Then, we give the evolution of some geometric structures of the manifold along this flow.
Generalized $\eta$-Ricci solitons on para-Kenmotsu manifolds associated to the Zamkovoy connection
Shahroud Azami 대한수학회 2024 대한수학회논문집 Vol.39 No.1
In this paper, we study para-Kenmotsu manifolds admitting generalized $\eta$-Ricci solitons associated to the Zamkovoy connection. We provide an example of generalized $\eta$-Ricci soliton on a para-Kenmotsu manifold to prove our results.
Evolution of the First Eigenvalue of Weighted p-Laplacian along the Yamabe Flow
Azami, Shahroud Department of Mathematics 2019 Kyungpook mathematical journal Vol.59 No.2
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.
RICCI 𝜌-SOLITONS ON 3-DIMENSIONAL 𝜂-EINSTEIN ALMOST KENMOTSU MANIFOLDS
Azami, Shahroud,Fasihi-Ramandi, Ghodratallah Korean Mathematical Society 2020 대한수학회논문집 Vol.35 No.2
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.
Azami, Shahroud Korean Mathematical Society 2020 대한수학회논문집 Vol.35 No.3
In this article we study the evolution and monotonicity of the first non-zero eigenvalue of weighted p-Laplacian operator which it acting on the space of functions on closed oriented Riemannian n-manifolds along the extended Ricci flow and normalized extended Ricci flow. We show that the first eigenvalue of weighted p-Laplacian operator diverges as t approaches to maximal existence time. Also, we obtain evolution formulas of the first eigenvalue of weighted p-Laplacian operator along the normalized extended Ricci flow and using it we find some monotone quantities along the normalized extended Ricci flow under the certain geometric conditions.
THE SET OF ZOLL METRICS IS NOT PRESERVED BY SOME GEOMETRIC FLOWS
Azami, Shahroud,Fasihi-Ramandi, Ghodratallah Korean Mathematical Society 2019 대한수학회논문집 Vol.34 No.3
The geodesics on the round 2-sphere $S^2$ are all simple closed curves of equal length. In 1903 Otto Zoll introduced other Riemannian surfaces with the same property. After that, his name is attached to the Riemannian manifolds whose geodesics are all simple closed curves of the same length. The question that "whether or not the set of Zoll metrics on 2-sphere $S^2$ is connected?" is still an outstanding open problem in the theory of Zoll manifolds. In the present paper, continuing the work of D. Jane for the case of the Ricci flow, we show that a naive application of some famous geometric flows does not work to answer this problem. In fact, we identify an attribute of Zoll manifolds and prove that along the geometric flows this quantity no longer reflects a Zoll metric. At the end, we will establish an alternative proof of this fact.
New volume comparison with almost Ricci soliton
Shahroud Azami,Sakineh Hajiaghasi 대한수학회 2022 대한수학회논문집 Vol.37 No.3
In this paper we consider a condition on the Ricci curvature involving vector fields which enabled us to achieve new results for volume comparison and Laplacian comparison. These results in special case obtained with considering volume non-collapsing condition. Also, by applying this condition we get new results of volume comparison for almost Ricci solitons.