Gradient Ricci Solitons Research Articles

Volume Growth Estimates for Ricci Solitons and Quasi-Einstein Manifolds

In this article, we provide some volume growth estimates for complete noncompact gradient Ricci solitons and quasi-Einstein manifolds similar to the classical results by Bishop, Calabi and Yau for complete Riemannian manifolds with nonnegative Ricci curvature. We prove a sharp volume growth estimate for complete noncompact gradient shrinking Ricci soliton. Moreover, we provide upper bound volume growth estimates for complete noncompact quasi-Einstein manifolds with $$\lambda =0.$$ In addition, we prove that geodesic balls of complete noncompact quasi-Einstein manifolds with $$\lambda <0$$ and $$\mu \le 0$$ have at most exponential volume growth.

Perfect fluid spacetimes and gradient solitons

The main object of this paper is to characterize the perfect fluid spacetimes if its metrics are Ricci solitons, gradient Ricci solitons, gradient A-Einstein solitons and gradient Schouten solitons.

Ricci almost solitons on semi‐Riemannian warped products

AbstractIt is shown that a gradient Ricci almost soliton on a warped product, whose potential function f depends on the fiber, is either a Ricci soliton or λ is not constant and the warped product, the base and the fiber are Einstein manifolds, which admit conformal vector fields. Assuming completeness, a classification is provided for the gradient Ricci almost solitons on warped products, whose potential functions depend on the fiber. An important decomposition property of the potential function in terms of functions which depend either on the base or on the fiber is proven. In the case of a complete gradient Ricci soliton, the potential function depends only on the base.

Clairaut Riemannian maps whose total manifolds admit a Ricci soliton

In this paper, we study Clairaut Riemannian maps whose total manifolds admit a Ricci soliton and give a nontrivial example of such Clairaut Riemannian maps. First, we calculate Ricci tensors and scalar curvature of total manifolds of Clairaut Riemannian maps. Then we obtain necessary conditions for the fibers of such Clairaut Riemannian maps to be Einstein and almost Ricci solitons. We also obtain a necessary condition for vector field [Formula: see text] to be conformal, where [Formula: see text] is a geodesic curve on total manifold of Clairaut Riemannian map. Further, we show that if total manifolds of Clairaut Riemannian maps admit a Ricci soliton with the potential mean curvature vector field of [Formula: see text] then the total manifolds of Clairaut Riemannian maps also admit a gradient Ricci soliton and obtain a necessary and sufficient condition for such maps to be harmonic by solving Poisson equation.

Geometry of semi-Riemannian group manifold and its applications in spacetime admitting Ricci solitons

In this work, we show that semi-Riemannian group manifold admits Ricci solitons and satisfies the dynamical cosmology equation of spacetime. In Sec. 2, we introduce and provide some geometric properties of semisymmetric nonmetric connection in semi-Riemannian space. In Sec. 3, we define and show some geometric properties of group manifold endowed with semisymmetric nonmetric connection in semi-Riemannian space. In the section that follows, we give a condition of a group manifold to be Ricci solitons and gradient Ricci soliton. In Sec. 5, we provide the applications of group manifold admitting Ricci solitons in the theory of general relativity.

$\eta$-RICCI SOLITONS AND GRADIENT RICCI SOLITONS ON $\delta$- LORENTZIAN TRANS-SASAKIAN MANIFOLDS

The objective of the present research article is to study the $\delta$-Lorentzian trans-Sasakian manifolds conceding the $\eta$-Ricci solitons and gradient Ricci soliton. We shown that a symmetric second order covariant tensor in a $\delta$-Lorentzian trans-Sasakian manifold is a constant multiple of metric tensor. Also, we furnish an example of $\eta$-Ricci soliton on 3-diemsional $\delta$-Lorentzian trans-Sasakian manifold is provide in the region where $\delta$-Lorentzian trans-Sasakian manifold is expanding. Furthermore, we discuss some results based on gradient Ricci solitons on $3$-dimensional $\delta$- Lorentzian trans-Sasakian manifold.

A note on gradient Ricci soliton warped metrics

AbstractIn this note, we prove triviality and nonexistence results for gradient Ricci soliton warped metrics. The proofs stem from the construction of gradient Ricci solitons that are realized as warped products, from which we know that the base spaces of these products are Ricci–Hessian type manifolds. We study this latter class of manifolds as the most appropriate setting to prove our results.

Curvature Estimates and Gap Theorems for Expanding Ricci Solitons

AbstractWe derive a sharp lower bound for the scalar curvature of non-flat and non-compact expanding gradient Ricci soliton provided that the scalar curvature is non-negative and the potential function is proper. Upper bound for the scalar curvature of expander with nonpositive Ricci curvature will also be given. Furthermore, we provide a sufficient condition for the scalar curvature of expanding soliton being non-negative. Curvature estimates of expanding solitons in dimensions three and four will also be established. As an application, we prove a gap theorem on 3D gradient expander.

Three-dimensional Ricci-degenerate Riemannian manifolds satisfying geometric equations

In this paper, we study a three-dimensional Ricci-degenerate Riemannian manifold \((M^3,g)\) that admits a smooth nontrivial solution f to the equation $$\begin{aligned} \nabla df=\psi Rc+\phi g, \end{aligned}$$ (1) where \(\psi ,\phi \) are given smooth functions of f, Rc is the Ricci tensor of g. Spaces of this type include various interesting classes, namely gradient Ricci solitons, m-quasi Einstein metrics, (vacuum) static spaces, V-static spaces, and critical point metrics. The m-quasi Einstein metrics and vacuum static spaces were previously studied in Jordan (Gen Relativ Gravit 41(9):2191–2280, 2009) and Kim and Shin (Math Nachr 292(8): 1727–1750, 2019), respectively. In this paper, we refine them and develop a general approach for the solutions of (1). We specify the shape of the metric g satisfying (1) when \(\nabla f\) is not a Ricci-eigen vector. Then we focus on the remaining three classes, namely gradient Ricci solitons, V-static spaces, and critical point metrics. Furthermore, we present classifications of local three-dimensional Ricci-degenerate spaces of these three classes by explicitly describing the metric g and the potential function f.

On generalized quasi Einstein standard static spacetimes

In the present paper, first, we characterize the standard static spacetimes satisfying certain Ricci-Hessian class type equations, such as generalized quasi Einstein manifolds, m-quasi Einstein manifolds, (m,ρ)-quasi Einstein manifolds, and gradient Ricci solitons. Then, we obtain some necessary and sufficient conditions about the existence of certain vector fields on the standard static spacetime. We also characterize the m-quasi Einstein standard static spacetimes admitting concurrent or parallel vector fields and get several physical properties of such spacetimes. At last, we construct non-trivial examples of m-quasi Einstein spacetimes verifying our results.

A Schwarz lemma and a Liouville theorem for generalized harmonic maps

When the sectional curvature of the target manifold is negative, we establish a Schwarz lemma for f-harmonic maps, if the dimension of the domain and the target is large, the result improves Theorem 3 in Chen and Zhao (2017) for the case of V=∇f. When the sectional curvature of the target is nonpositive, we obtain a Liouville theorem for the general V-harmonic maps, as a consequence, any V-harmonic function u, satisfying |u(x)|=o(r(x)), on a complete Riemannian manifold with nonnegative Bakry–Emery–Ricci curvature is a constant. We also give some applications on gradient Ricci solitons and gradient solitons with potential which are solutions to Ricci-harmonic flow.

Conformal Ricci soliton and quasi-Yamabe soliton on generalized Sasakian space form

The present paper is devoted to generalized Sasakian space forms admitting conformal Ricci soliton and Quasi-Yamabe soliton. Nature of the conformal Ricci soliton is characterized on generalized Sasakian space form with various types of the potential vector field, and conditions for the conformal Ricci soliton to be shrinking, steady, or expanding are also given. Then it is shown that, depending on the nature of the structure functions of a generalized Sasakian space form, the potential function of a conformal gradient Ricci soliton is constant. Next, it is proved that under certain conditions, a quasi-Yamabe soliton reduces to a Yamabe soliton on generalized Sasakian space forms. Finally, an illustrative example of a generalized Sasakian space form is discussed to verify our results.

Perelman-type no breather theorem for noncompact Ricci flows

In this paper, we first show that a complete shrinking breather with Ricci curvature bounded from below must be a shrinking gradient Ricci soliton. This result has several applications. First, we can classify all complete $3$-dimensional shrinking breathers. Second, we can show that every complete shrinking Ricci soliton with Ricci curvature bounded from below must be gradient -- a generalization of Naber's result. Furthermore, we develop a general condition for the existence of the asymptotic shrinking gradient Ricci soliton, which hopefully will contribute to the study of ancient solutions.

Four-dimensional complete gradient shrinking Ricci solitons

Abstract In this article, we study four-dimensional complete gradient shrinking Ricci solitons. We prove that a four-dimensional complete gradient shrinking Ricci soliton satisfying a pointwise condition involving either the self-dual or anti-self-dual part of the Weyl tensor is either Einstein, or a finite quotient of either the Gaussian shrinking soliton ℝ 4 {\mathbb{R}^{4}} , or 𝕊 3 × ℝ {\mathbb{S}^{3}\times\mathbb{R}} , or 𝕊 2 × ℝ 2 . {\mathbb{S}^{2}\times\mathbb{R}^{2}.} In addition, we provide some curvature estimates for four-dimensional complete gradient Ricci solitons assuming that its scalar curvature is suitable bounded by the potential function.

Gradient Estimates for a Class of Semilinear Parabolic Equations and Their Applications

In this paper we address the following parabolic equation @@ on a smooth metric measure space with Bakry–Emery curvature bounded from below for F being a differentiable function defined on $\mathbb {R}$ . Our motivation is originally inspired by gradient estimates of Allen–Cahn and Fisher-KKP equations (Bǎilesteanu, M., Ann. Glob. Anal. Geom. 51, 367–378, 2017; Cao et al., Pac. J. Math. 290, 273–300, 2017). We show new gradient estimates for these equations. As their applications, we obtain Liouville type theorems for positive or bounded solutions to the above equation when either F = cu(1 − u) (the Fisher-KKP equation) or; F = −u3 + u (the Allen–Cahn equation); or $F=au\log u$ (the equation involving gradient Ricci solitons).

On Ricci curvature of metric structures on [formula omitted]-manifolds

We study the properties of Ricci curvature of g-manifolds with particular attention paid to higher dimensional abelian Lie algebra case. The relations between Ricci curvature of the manifold and the Ricci curvature of the transverse manifold of the characteristic foliation are investigated. In particular, sufficient conditions are found under which the g-manifold can be a Ricci soliton or a gradient Ricci soliton. Finally, we obtain a amazing (non-existence) higher dimensional generalization of the Boyer-Galicki theorem on Einstein K-manifolds for a special class of abelian g-manifolds.

Three-dimensional Riemannian manifolds and Ricci solitons

We characterize the three-dimensional Riemannian manifolds endowed with a semi-symmetric metric ρ-connection if its Riemannian metrics are Ricci and gradient Ricci solitons, respectively. It is proved that if a three-dimensional Riemannian manifold equipped with a semi-symmetric metric ρ-connection admits a Ricci soliton, then the manifold possesses the constant sectional curvature −1 and the soliton is expanding with λ = −2. Next, we study the gradient Ricci solitons in such a manifold. Finally, we construct a non-trivial example of a three-dimensional Riemannian manifold endowed with a semi-symmetric metric ρ-connection admitting a Ricci soliton and validate our some results.

η-Ricci Solitons and Gradient Ricci Solitons on f-Kenmotsu Manifolds

The aim of the present research article is to study the f-kenmotsu manifolds admitting the η-Ricci Solitons and gradient Ricci solitons with respect to the semi-symmetric non metric connection.

A note on almost Ricci soliton and gradient almost Ricci soliton on para-Sasakian manifolds

The object of the offering exposition is to study almost Ricci soliton and gradient almost Ricci soliton in 3-dimensional para-Sasakian manifolds. At first, it is shown that if $(g, V,\lambda)$ be an almost Ricci soliton on a 3-dimensional para-Sasakian manifold $M$, then it reduces to a Ricci soliton and the soliton is expanding for $\lambda$=-2. Besides these, in this section, we prove that if $V$ is pointwise collinear with $\xi$, then $V$ is a constant multiple of $\xi$ and the manifold is of constant sectional curvature $-1$. Moreover, it is proved that if a 3-dimensional para-Sasakian manifold admits gradient almost Ricci soliton under certain conditions then either the manifold is of constant sectional curvature $-1$ or it reduces to a gradient Ricci soliton. Finally, we consider an example to justify some results of our paper.

Certain solitons on generalized $(\kappa, \mu)$ contact metric manifolds

The aim of the present paper is to study some solitons on three dimensional generalized $(\kappa, \mu)$-contact metric manifolds. We study gradient Yamabe solitons on three dimensional generalized $(\kappa, \mu)$-contact metric manifolds. It is proved that if the metric of a three dimensional generalized $(\kappa, \mu)$-contact metric manifold is gradient Einstein soliton then $\mu = \frac{2\kappa}{\kappa - 2}.$ It is shown that if the metric of a three dimensional generalized $(\kappa, \mu)$-contact metric manifold is closed m-quasi Einstein metric then $\kappa = \frac{\lambda}{m + 2}$ and $\mu = 0.$ We also study conformal gradient Ricci solitons on three dimensional generalized $(\kappa, \mu)$-contact metric manifolds.