Let M be an n-dimensional closed Riemannian manifold with metric g, dμ = e−φ(x)dν be the weighted... more Let M be an n-dimensional closed Riemannian manifold with metric g, dμ = e−φ(x)dν be the weighted measure and ∆p,φ 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.
International Journal of Geometric Methods in Modern Physics, Apr 27, 2022
Let [Formula: see text], [Formula: see text], be a compact Finsler manifold. In this paper, we co... more Let [Formula: see text], [Formula: see text], be a compact Finsler manifold. In this paper, we consider a Finsler manifold evolving by the Finsler-geometric flow [Formula: see text], where [Formula: see text] is the symmetric metric tensor associated with [Formula: see text], and [Formula: see text] is a symmetric [Formula: see text]-tensor, and the study parabolic equation [Formula: see text] where [Formula: see text] is a function on [Formula: see text] of [Formula: see text] in [Formula: see text]-variables and [Formula: see text] in [Formula: see text]-variable and [Formula: see text] is a function of [Formula: see text] in [Formula: see text]. We obtain a global estimate and a Harnack estimate for positive solutions. Our results are also natural extension of similar results on Riemannian-geometric flow.
Let (Mn , F(t), m), t ∈ [0, T], be a compact Finsler manifold with F(t) evolving by the Finsler-g... more Let (Mn , F(t), m), t ∈ [0, T], be a compact Finsler manifold with F(t) evolving by the Finsler-geometric flow $\begin{array}{} \displaystyle \frac{\partial g(x,t)}{\partial t}=2h(x,t), \end{array}$ where g(t) is the symmetric metric tensor associated with F, and h(t) is a symmetric (0, 2)-tensor. In this paper, we consider local Li-Yau type gradient estimates for positive solutions of the following nonlinear heat equation with potential $$\begin{array}{} \displaystyle \partial_{t}u(x,t)=\Delta_{m}u(x,t)-\mathcal{R}(x,t)u(x,t) -au(x,t)\log u(x,t),\quad(x,t)\in M\times [0,T], \end{array}$$ along the Finsler-geometric flow, where 𝓡 is a smooth function, and a is a real nonpositive constant. As an application we obtain a global estimate and a Harnack estimate. Our results are also natural extension of similar results on Riemannian-geometric flow.
Asian-european Journal of Mathematics, Nov 19, 2022
We investigate the behavior of the lowest geometric constant, [Formula: see text], along the exte... more We investigate the behavior of the lowest geometric constant, [Formula: see text], along the extended Ricci flow such that there exist positive solutions to the following partial differential equation: [Formula: see text] with [Formula: see text], where [Formula: see text] and [Formula: see text] are real constants. We drive the evolution formula for the geometric constant [Formula: see text] along the unnormalized and normalized extended Ricci flow. Moreover, we give some monotonic quantities involving [Formula: see text] along the extended Ricci flow by imposing some geometric conditions.
In this paper, we will focus our attention on the structure of h-almost Ricci solitons. A complet... more In this paper, we will focus our attention on the structure of h-almost Ricci solitons. A complete classification of h-almost Ricci solitons with concurrent potential vector fields is given. Also, we obtain conditions on a submanifold of a Riemannian h-almost Ricci soliton to be an h-almost Ricci soliton. Finally, we classify h-almost Ricci soliton on Euclidean hypersurface with λ = h.
Arab Journal of Mathematical Sciences, Aug 31, 2021
In this paper, we investigate the geometry of left-invariant Randers metrics on the Heisenberg gr... more In this paper, we investigate the geometry of left-invariant Randers metrics on the Heisenberg group H 2n+1 , of dimension 2n + 1. Considering a left-invariant Randers metric, we give the Levi-Civita connection, curvature tensor, Ricci tensor and scalar curvature and show the Heisenberg groups H 2n+1 have constant negative scalar curvature. Also, we show the Heisenberg group H 2n+1 can not admit Randers metric of Berwald and Ricci-quadratic Douglas types. Finally, an explicit formula for computing flag curvature is obtained which shows that there exist flags of strictly negative and strictly positive curvatures.
Let M be an n-dimensional closed Riemannian manifold with metric g, dμ = e−φ(x)dν be the weighted... more Let M be an n-dimensional closed Riemannian manifold with metric g, dμ = e−φ(x)dν be the weighted measure and ∆p,φ 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.
International Journal of Geometric Methods in Modern Physics, Apr 27, 2022
Let [Formula: see text], [Formula: see text], be a compact Finsler manifold. In this paper, we co... more Let [Formula: see text], [Formula: see text], be a compact Finsler manifold. In this paper, we consider a Finsler manifold evolving by the Finsler-geometric flow [Formula: see text], where [Formula: see text] is the symmetric metric tensor associated with [Formula: see text], and [Formula: see text] is a symmetric [Formula: see text]-tensor, and the study parabolic equation [Formula: see text] where [Formula: see text] is a function on [Formula: see text] of [Formula: see text] in [Formula: see text]-variables and [Formula: see text] in [Formula: see text]-variable and [Formula: see text] is a function of [Formula: see text] in [Formula: see text]. We obtain a global estimate and a Harnack estimate for positive solutions. Our results are also natural extension of similar results on Riemannian-geometric flow.
Let (Mn , F(t), m), t ∈ [0, T], be a compact Finsler manifold with F(t) evolving by the Finsler-g... more Let (Mn , F(t), m), t ∈ [0, T], be a compact Finsler manifold with F(t) evolving by the Finsler-geometric flow $\begin{array}{} \displaystyle \frac{\partial g(x,t)}{\partial t}=2h(x,t), \end{array}$ where g(t) is the symmetric metric tensor associated with F, and h(t) is a symmetric (0, 2)-tensor. In this paper, we consider local Li-Yau type gradient estimates for positive solutions of the following nonlinear heat equation with potential $$\begin{array}{} \displaystyle \partial_{t}u(x,t)=\Delta_{m}u(x,t)-\mathcal{R}(x,t)u(x,t) -au(x,t)\log u(x,t),\quad(x,t)\in M\times [0,T], \end{array}$$ along the Finsler-geometric flow, where 𝓡 is a smooth function, and a is a real nonpositive constant. As an application we obtain a global estimate and a Harnack estimate. Our results are also natural extension of similar results on Riemannian-geometric flow.
Asian-european Journal of Mathematics, Nov 19, 2022
We investigate the behavior of the lowest geometric constant, [Formula: see text], along the exte... more We investigate the behavior of the lowest geometric constant, [Formula: see text], along the extended Ricci flow such that there exist positive solutions to the following partial differential equation: [Formula: see text] with [Formula: see text], where [Formula: see text] and [Formula: see text] are real constants. We drive the evolution formula for the geometric constant [Formula: see text] along the unnormalized and normalized extended Ricci flow. Moreover, we give some monotonic quantities involving [Formula: see text] along the extended Ricci flow by imposing some geometric conditions.
In this paper, we will focus our attention on the structure of h-almost Ricci solitons. A complet... more In this paper, we will focus our attention on the structure of h-almost Ricci solitons. A complete classification of h-almost Ricci solitons with concurrent potential vector fields is given. Also, we obtain conditions on a submanifold of a Riemannian h-almost Ricci soliton to be an h-almost Ricci soliton. Finally, we classify h-almost Ricci soliton on Euclidean hypersurface with λ = h.
Arab Journal of Mathematical Sciences, Aug 31, 2021
In this paper, we investigate the geometry of left-invariant Randers metrics on the Heisenberg gr... more In this paper, we investigate the geometry of left-invariant Randers metrics on the Heisenberg group H 2n+1 , of dimension 2n + 1. Considering a left-invariant Randers metric, we give the Levi-Civita connection, curvature tensor, Ricci tensor and scalar curvature and show the Heisenberg groups H 2n+1 have constant negative scalar curvature. Also, we show the Heisenberg group H 2n+1 can not admit Randers metric of Berwald and Ricci-quadratic Douglas types. Finally, an explicit formula for computing flag curvature is obtained which shows that there exist flags of strictly negative and strictly positive curvatures.
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Papers by Shahroud Azami