Gradient Ricci Solitons Research Articles

Classification of superpotentials for cohomogeneity one Ricci solitons

We classify superpotentials for the Hamiltonian system corresponding to the cohomogeneity one gradient Ricci soliton equations. Aside from recovering known examples of superpotentials for steady solitons, we find a new superpotential on a specific case of the Bérard Bergery–Calabi ansatz. The latter is used to obtain an explicit formula for a steady complete soliton with an equidistant family of hypersurfaces given by circle bundles over S2×S2. There are no superpotentials in the non-steady case in dimensions greater than 2, even if polynomial coefficients are allowed. We also briefly discuss generalised first integrals and the limitations of some known methods of finding them.

Inhomogeneous deformations of Einstein solvmanifolds

AbstractFor each non‐flat, unimodular Ricci soliton solvmanifold , we construct a one‐parameter family of complete, expanding, gradient Ricci solitons that admit a cohomogeneity one isometric action by . The orbits of this action are hypersurfaces homothetic to . These metrics are asymptotic at one end to an Einstein solvmanifold. In the one‐parameter family, exactly one metric is Einstein, and exactly one has orbits that are isometric to .

On noncompact warped product Ricci solitons

AbstractThe goal of this paper is to investigate complete noncompact warped product gradient Ricci solitons. Nonexistence results, estimates for the warping function and for its gradient are proven. When the soliton is steady or expanding these nonexistence results generalize to a broader context certain estimates and rigidity obtained when studying warped product Einstein manifolds. When the soliton is shrinking, it is presented as a nonexistence theorem with no counterpart in the Einstein case, which is proved using properties of the first eigenvalue of a weighted Laplacian.

When are shrinking gradient Ricci soliton compact

Suppose (M4,g,f) is a complete shrinking gradient Ricci soliton. We give a sufficient condition for a soliton to be compact, generalizing previous result of Munteanu-Wang [17]. As an application, we give a classification of (M4,g,f) under some natural conditions.

Investigations on a Riemannian Manifold with a Semi- Symmetric Non-Metric Connection and Gradient Solitons

This article carries out the investigation of a three-dimensional Riemannian manifold N3 endowed with a semi-symmetric type non-metric connection. Firstly, we construct a non-trivial example to prove the existence of a semi-symmetric type non-metric connection on N3. It is established that a N3 with the semi-symmetric type non-metric connection, whose metric is a gradient Ricci soliton, is a manifold of constant sectional curvature with respect to the semi-symmetric type non-metric connection. Moreover, we prove that if the Riemannian metric of N3 with the semi-symmetric type non-metric connection is a gradient Yamabe soliton, then either N3 is a manifold of constant scalar curvature or the gradient Yamabe soliton is trivial with respect to the semi-symmetric type non-metric connection. We also characterize the manifold N3 with a semi-symmetric type non-metric connection whose metrics are Einstein solitons and m-quasi Einstein solitons of gradient type, respectively.

Geometry and analysis of gradient Ricci solitons in dimension four

Geometry and analysis of gradient Ricci solitons in dimension four

Lorentzian manifolds: A characterization with a type of semi-symmetric non-metric connection

This paper characterizes the Lorentzian manifolds endowed with a semi-symmetric non-metric [Formula: see text]-connection (briefly, ssnmρc). First, the existence of semi-symmetric non-metric connection (ssnmc) on Lorentzian manifold is established, and it is shown that an [Formula: see text]-dimensional Lorentzian manifold equipped with an ssnmρc is a generalized Robertson–Walker spacetime. We also establish the condition for a Lorentzian manifold together with an ssnmρc to be a Robertson–Walker spacetime. In this series, the properties of Ricci semi-symmetric Lorentzian manifold endowed with an ssnmρc, almost Ricci solitons and gradient Ricci solitons are explored. Finally, a non-trivial example of Lorentzian manifold admits an ssnmρc, which is constructed to verify some of our results.

A characterization of Ricci solitons in Lorentzian generalized Sasakian space forms

In this paper, we study Lorentzian generalized Sasakian space forms admitting Ricci soliton, conformal gradient Ricci soliton and Ricci Yamabe soliton. We also study the conditions for the soliton to be steady, shrinking and expanding. We also provide some applications of Ricci Yamabe solitons.

Gradient Ricci solitons on Kropina measure spaces

Let F=α2β be a Kropina metric on a measure space (M,m) with the navigation data (h,W). In this paper, we give an equivalent characterization for (M,F,m) to be a gradient almost Ricci soliton, which implies that every gradient almost Ricci soliton has vanishing SBH-curvature. Based on this, we prove that (M,F,m) is a gradient almost Ricci soliton if and only if (M,h,f) is a Riemannian gradient almost Ricci soliton, W is a unit Killing vector field on M and f satisfies a differential equation, where f is the potential function f of the measure m (see Theorem 1.2). As an application, we construct some new shrinking, steady and expanding gradient Ricci solitons.

Existence of Ricci soliton vector fields on Vaidya spacetime

In this paper, we investigate the behaviour of the Vaidya spacetime admitting a Ricci soliton vector field, where we have found the expressions for the four components of the vector field, and we have shown, under this condition, that the spacetime reduces to the Schwarzschild spacetime. Moreover, we have found the desired expression for the potential function h to satisfy in order for a gradient Ricci soliton vector field to exist on the Vaidya spacetime.

Improved oscillation estimates and the Hitchin–Thorpe inequality on compact Ricci solitons

Stimulated by improved oscillation estimates of the potential function and the scalar curvature on compact gradient Ricci solitons introduced in a recent study by Cheng, Ribeiro, and Zhou [Proc. Am. Math. Soc. Ser. B 10, 33–45 (2023)], we give several new sufficient conditions for compact four-dimensional normalized shrinking Ricci solitons to satisfy the Hitchin–Thorpe inequality. Our new conditions refine the validity of the Hitchin–Thorpe inequality obtained by Tadano [J. Math. Phys. 58, 023503 (2017)], Tadano [J. Math. Phys. 59, 043507 (2018)], and Tadano [Differ. Geom. Appl. 66, 231–241 (2019)].

On harmonic and biharmonic maps from gradient Ricci solitons

AbstractWe study harmonic and biharmonic maps from gradient Ricci solitons. We derive a number of analytic and geometric conditions under which harmonic maps are constant and which force biharmonic maps to be harmonic. In particular, we show that biharmonic maps of finite energy from the two‐dimensional cigar soliton must be harmonic.

Gradient pseudo‐Ricci solitons of real hypersurfaces

AbstractLet M be a real hypersurface of a complex space form , . Suppose that the structure vector field ξ of M is an eigen vector field of the Ricci tensor S, , β being a function. We study on M, a gradient pseudo‐Ricci soliton () that is an extended concept of gradient Ricci soliton, closely related to pseudo‐Einstein real hypersurfaces. When , we show that M is a Hopf hypersurface.

Ricci solitons on general relativistic spacetimes

The main aim of this manuscript is to characterize the general relativistic spacetimes with Ricci and gradient Ricci solitons. It is proven that if the metric of a general relativistic spacetime (M 4, ξ) admitting a special unit timelike vector field ξ is an almost Ricci soliton (g, ξ, λ), then (M 4, ξ) is a perfect fluid spacetime, and almost Ricci soliton (g, ξ, λ) on (M 4, ξ) becomes shrinking Ricci soliton. We prove that a general relativistic perfect fluid spacetime equipped with a special unit timelike vector field together with a Ricci soliton is an Einstein spacetime. In this sequel, we also prove that the Ricci soliton is shrinking, soliton vector field is Killing and the scalar curvature of the perfect fluid spacetime is constant. It is proven that a general relativistic perfect fluid spacetime together with a Ricci soliton is a generalized Robertson-Walker (GRW) spacetime. The existence of gradient Ricci solitons on general relativistic spacetimes are established. We also construct a non-trivial example of general relativistic spacetime equipped with a special unit timelike vector field, and verify some of our theorems.

Ricci Soliton of CR-Warped Product Manifolds and Their Classifications

In this article, we derived an equality for CR-warped product in a complex space form which forms the relationship between the gradient and Laplacian of the warping function and second fundamental form. We derived the necessary conditions of a CR-warped product submanifolds in Ka¨hler manifold to be an Einstein manifold in the impact of gradient Ricci soliton. Some classification of CR-warped product submanifolds in the Ka¨hler manifold by using the Euler–Lagrange equation, Dirichlet energy and Hamiltonian is given. We also derive some characterizations of Einstein warped product manifolds under the impact of Ricci Curvature and Divergence of Hessian tensor.

On Bochner Flat Kähler B-Manifolds

We obtain on a Kähler B-manifold (i.e., a Kähler manifold with a Norden metric) some corresponding results from the Kählerian and para-Kählerian context concerning the Bochner curvature. We prove that such a manifold is of constant totally real sectional curvatures if and only if it is a holomorphic Einstein, Bochner flat manifold. Moreover, we provide the necessary and sufficient conditions for a gradient Ricci soliton or a holomorphic η-Einstein Kähler manifold with a Norden metric to be Bochner flat. Finally, we show that a Kähler B-manifold is of quasi-constant totally real sectional curvatures if and only if it is a holomorphic η-Einstein, Bochner flat manifold.

The entropy of Ricci flows with Type-I scalar curvature bounds

In this paper, we extend the theory of Ricci flows satisfying a Type-I scalar curvature bound at a finite-time singularity. In [2], Bamler showed that a Type-I rescaling procedure will produce a singular shrinking gradient Ricci soliton with singularities of codimension 4. We prove that the entropy of a conjugate heat kernel based at the singular time converges to the soliton entropy of the singular soliton, and use this to characterize the singular set of the Ricci flow solution in terms of a heat kernel density function. This generalizes results previously only known with the stronger assumption of a Type-I curvature bound. We also show that in dimension 4, the singular Ricci soliton is smooth away from finitely many points, which are conical smooth orbifold singularities.

On Euler characteristic and Hitchin-Thorpe inequality for four-dimensional compact Ricci solitons

In this article, we investigate the geometry of 4 4 -dimensional compact gradient Ricci solitons. We prove that, under an upper bound condition on the range of the potential function, a 4 4 -dimensional compact gradient Ricci soliton must satisfy the classical Hitchin-Thorpe inequality. In addition, some volume estimates are also obtained.

An Optimal Volume Growth Estimate for Noncollapsed Steady Gradient Ricci Solitons

In this paper, we prove a volume growth estimate for steady gradient Ricci solitons with bounded Nash entropy. We show that such a steady gradient Ricci soliton has volume growth rate no smaller than $$r^{\frac{n+1}{2}}.$$ This result not only improves the estimate in (Chan et al., arXiv:2107.01419 , 2021, Theorem 1.3), but also is optimal since the Bryant soliton and Appleton’s solitons (Appleton, arXiv:1708.00161 , 2017) have exactly this growth rate.

Almost Kenmotsu Manifolds

The object of this paper is to study generalized φ-recurrent almost Kenmotsu manifolds with characteristic vector field ξ belonging to (k, µ)-nullity distribution. We have showed that these manifolds reduce to Kenmotsu manifolds with scalar curvature-1. Further we establish the relations among the associated 1-forms and proved the conditions under which gradient Ricci almost soliton reduce to gradient Ricci soliton.