Expanding Ricci Solitons Research Articles

Ricci solitons on anisotropic fluid spacetime in f(R, T)-gravity

In this paper we study the behaviour of anisotropic fluid spacetimes in f(R, T)-gravity for admitting Ricci Soliton. We obtain the form of tangential pressure of the fluid for steady, shrinking and expanding Ricci soliton when Ricci soliton vector field becomes a Killing vector field. Modified Poisson and Liouville equations are derived from the Ricci soliton equation in f(R, T)-gravity having anisotropic fluid. Also harmonic significance of Ricci soliton on anisotropic fluid spacetime in f(R, T)-gravity with a harmonic potential function is derived. We derive field equations for perfect fluid plane symmetric static spacetime and obtain their solutions. For each solution we then obtain Ricci soliton vector fields. We observe that such spacetimes admit expanding, shrinking or steady Ricci solitons. All the obtained solitons are trivial in nature. When these Ricci soliton vector fields become Killing vector fields then our results are in full agreement with the theorems deduced from our study.

Rigidity Results for Shrinking and Expanding Ricci Solitons

Rigidity Results for Shrinking and Expanding Ricci Solitons

Ricci solitons on pseudo Riemannian generalized symmetric spaces

We study the geometry of four-dimensional pseudo Riemannian generalized symmetric spaces of type D; whose metric was explicitly described by Cerny and Kowalski. After describing their curvature properties; we classify the Killing vectors field of these spaces and more particularly, we study the existence of non-trivial (i.e., not Einstein) Ricci solitons; we show that these spaces are shrinking or expanding Ricci solitons but never steady. Moreover this Ricci soliton is not a gradient one.

On the classification of four-dimensional gradient Ricci solitons

In this paper, we prove some classification results for four-dimensional gradient Ricci solitons. For a four-dimensional gradient shrinking Ricci soliton with div4Rm±=0, we show that it is either Einstein or a finite quotient of R4, S2×R2 or S3×R. The same result can be obtained under the condition of div4W±=0. We also present some classification results of four-dimensional complete non-compact gradient expanding Ricci soliton with non-negative Ricci curvature and gradient steady Ricci solitons under certain curvature conditions.

Curvature estimates for 4-dimensional complete gradient expanding Ricci solitons

Abstract In this paper, we derive curvature estimates for 4-dimensional complete gradient expanding Ricci solitons with nonnegative Ricci curvature (outside a compact set K). More precisely, we prove that the norm of the curvature tensor Rm {\mathrm{{Rm}}} and its covariant derivative ∇ ⁡ Rm {\nabla\mathrm{{Rm}}} can be bounded by the scalar curvature R by | Rm | ≤ C a ⁢ R a {|\mathrm{{Rm}}|\leq C_{a}R^{a}} and | ∇ ⁡ Rm | ≤ C a ⁢ R a {|\nabla\mathrm{{Rm}}|\leq C_{a}R^{a}} (on M ∖ K {M\setminus K} ), for any 0 ≤ a < 1 {0\leq a<1} and some constant C a > 0 {C_{a}>0} . Moreover, if the scalar curvature has at most polynomial decay at infinity, then | Rm | ≤ C ⁢ R {|\mathrm{{Rm}}|\leq CR} (on M ∖ K {M\setminus K} ). As an application, it follows that if a 4-dimensional complete gradient expanding Ricci soliton ( M 4 , g , f ) {(M^{4},g,f)} has nonnegative Ricci curvature and finite asymptotic scalar curvature ratio then it has finite asymptotic curvature ratio, hence admits C 1 , α {C^{1,\alpha}} asymptotic cones at infinity ( 0 < α < 1 ) {(0<\alpha<1)} according to Chen and Deruelle (2015).[21].

The classification of Lorentzian Lie groups with non‐Killing left‐invariant conformal vector fields

This paper is to classify Lorentzian Lie groups ( G , ⟨ · , · ⟩ ) $(G,\langle \cdot ,\cdot \rangle )$ admitting non-Killing left-invariant conformal vector fields whose Lie algebra is g ${\mathfrak {g}}$ . As we know, g ${\mathfrak {g}}$ is solvable and [ g , g ] $[{\mathfrak {g}},{\mathfrak {g}}]$ is of codimension 1 in g ${\mathfrak {g}}$ . In this paper, we will prove that [ g , g ] $[{\mathfrak {g}},{\mathfrak {g}}]$ is an Abelian Lie algebra, or a direct sum of a generalized Heisenberg Lie algebra and an Abelian Lie algebra. We obtain a simple criterion for such Lorentzian Lie groups with dim G ⩾ 4 $\dim G\geqslant 4$ to be conformally flat, and moreover, examples of conformally flat and non-conformally flat Lorentzian Lie groups are constructed. Finally, we prove that Lorentzian Lie groups admitting non-Killing left-invariant conformal vector fields can be shrinking, steady and expanding Ricci solitons.

Exact solutions of Diffusion Equation on sphere

‎We examine the diffusion‎ ‎equation on the sphere‎. ‎In this sense‎, ‎we answer question of the symmetry classification‎. ‎We provide the algebra of symmetry and build‎ ‎the optimal system of Lie subalgebras‎. ‎We prove for one-dimensional optimal systems of Eq‎.(4), ‎space is expanding Ricci solitons‎. ‎Reductions of similarities related to subalgebras are classified‎, ‎and some exact invariant solutions of the diffusion‎ ‎equation on the sphere are presented‎.

Ricci soliton vector fields of spherically symmetric static spacetimes

In this paper, we focused our attention to find Ricci solitons of spherically symmetric static spacetimes. It is shown that special classes of such spacetime metrics admit shrinking, expanding or steady Ricci solitons. It is found that the dimensions of the Ricci soliton vector fields are four, five, eight, ten or eleven. It is observed that non-Einstein metrics also exist with shrinking or expanding Ricci soliton vector fields. In all the cases, the obtained Ricci soliton vector fields are proved to be gradient and concircular potential fields.

Ricci almost soliton and almost Yamabe soliton on Kenmotsu manifold

We prove that a Ricci almost soliton on a Kenmotsu manifold of dimension [Formula: see text] reduces to an expanding Ricci soliton satifying certain condition on the potential vector field or on the soliton function. Next, we show that any Ricci almost soliton on a Kenmotsu manifold is trivial (Einstein) if the soliton vector leaves the contact form [Formula: see text] invariant. Finally, we classify (locally) a Kenmotsu manifold admitting an almost Yamabe soliton. Some examples have been constructed of almost Yamabe solitons on different class of warped product spaces.

Complete Ricci Solitons via Estimates on the Soliton Potential

Abstract In this paper, a growth estimate on the soliton potential is shown for a large class of cohomogeneity one manifolds. This is used to construct continuous families of complete steady and expanding Ricci solitons in the setups of Lü–Page–Pope [24] and Dancer–Wang [17]. It also provides a different approach to the two summands system [30] that applies to all known geometric examples.

Immortal homogeneous Ricci flows

We show that for an immortal homogeneous Ricci flow solution any sequence of parabolic blow-downs subconverges to a homogeneous expanding Ricci soliton. This is established by constructing a new Lyapunov function based on curvature estimates which come from real geometric invariant theory.

RIGIDITY OF GRADIENT SHRINKING AND EXPANDING RICCI SOLITONS

In this paper, we prove that a gradient shrinking Ricci soliton is rigid if the radial curvature vanishes and the second order divergence of Bach tensor is non-positive. Moreover, we show that a complete non-compact gradient expanding Ricci soliton is rigid if the radial curvature vanishes, the Ricci curvature is nonnegative and the second order divergence of Bach tensor is nonnegative.

Note on asymptotically conical expanding Ricci solitons

We show that at the level of formal expansions, any compact Riemannian manifold is the sphere at infinity of an asymptotically conical gradient expanding Ricci soliton.

Non-Kähler expanding Ricci solitons, Einstein metrics, and exotic cone structures

We produce new non-K\"ahler, non-Einstein, complete expanding gradient Ricci solitons with conical asymptotics and underlying manifold of the form $\R^2 \times M_2 \times \cdots \times M_r$, where $r \geq 2$ and $M_i$ are arbitrary closed Einstein spaces with positive scalar curvature. We also find numerical evidence for complete expanding solitons on the vector bundles whose sphere bundles are the twistor or ${\rm Sp}(1)$ bundles over quaternionic projective space.

Structure of homogeneous Ricci solitons and the Alekseevskii conjecture

We bring new insights into the longstanding Alekseevskii conjecture, namely that any connected homogeneous Einstein manifold of negative scalar curvature is diffeomorphic to a Euclidean space, by proving structural results which are actually valid for any homogeneous expanding Ricci soliton, and generalize many well-known results on Einstein solvmanifolds, solvsolitons, and nilsolitons. We obtain that any homogeneous expanding Ricci soliton $M = G/K$ is diffeomorphic to a product $U / K \times N$, where $U$ is a maximal reductive Lie subgroup of $G$ and $N$ is the maximal nilpotent normal subgroup of $G$, such that the metric restricted to $N$ is a nilsoliton. Moreover, strong compatibility conditions between the metric and the action of $U$ on $N$ by conjugation must hold, including a nice formula for the Ricci operator of the metric restricted to $U / K$. Our main tools come from geometric invariant theory. As an application, we give many Lie theoretical characterizations of algebraic solitons, as well as a proof of the fact that the following a priori much stronger result is actually equivalent to Alekseevskii’s conjecture: Any expanding algebraic soliton is diffeomorphic to a Euclidean space.

Two-Dimensional Gradient Ricci Solitons Revisited

In this note, we complete the classification of the geometry of non-compact two-dimensional gradient Ricci solitons. As a consequence, we obtain two corollaries: First, a complete two-dimensional gradient Ricci soliton has bounded curvature. Second, we give examples of complete two-dimensional expanding Ricci solitons with negative curvature that are topologically disks and are not hyperbolic space.

The canonical expanding soliton and Harnack inequalities for Ricci flow

We introduce the notion of Canonical Expanding Ricci Soliton, and use it to derive new Harnack inequalities for Ricci flow. This viewpoint also gives geometric insight into the existing Harnack inequalities of Hamilton and Brendle.

The second variation of the Ricci expander entropy

The critical points of the W + functional introduced by M. Feldman, T. Ilmanen and L. Ni are the expanding Ricci solitons, which are special solutions of the Ricci flow. On compact manifolds, expanding solitons coincide with Einstein metrics. In this paper, we compute the first and second variations of the entropy functional of the W + functional, and briefly discuss the linear stability of compact hyperbolic space forms.

On a sharp volume estimate for gradient Ricci solitons with scalar curvature bounded below

In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota's argument we obtain a local lower bound estimate of the scalar curvature for the Ricci flow on complete manifolds. Consequently, one has a sharp estimate of the scalar curvature for expanding Ricci solitons; we also provide a direct (elliptic) proof of this sharp estimate. Moreover, if the scalar curvature attains its minimum value at some point, then the manifold is Einstein.

Expanding Ricci solitons with pinched Ricci curvature

In this paper, we prove that there is no expanding gradient Ricci soliton with (positively) pinched Ricci curvature in dimension three. The same result is true for higher dimensions with the extra decay condition about the full curvature.