Ricci-harmonic Solitons Research Articles

On gradient normalized Ricci-harmonic solitons in sequential warped products

<p>Our investigation involved sequentially warped product manifolds that contained gradient-normalized Ricci-harmonic solitons. We presented the primary connections for a gradient-normalized Ricci-harmonic soliton on sequential warped product manifolds. In practical applications, our research investigated gradient-normalized Ricci-harmonic solitons for sequential generalized Robertson-Walker spacetimes and sequential standard static space-times. Our finding generalized all results proven in <sup>[<xref ref-type="bibr" rid="b26">26</xref>]</sup>.</p>

On sequential warped product manifolds admitting gradient Ricci-harmonic solitons

We consider sequential warped product manifolds admitting gradient Ricci-harmonic solitons. We give the main relations for a gradient Ricci-harmonic soliton on sequential warped product manifolds. As physical applications, we consider gradient Ricci-harmonic solitons on sequential generalized Robertson-Walker space-times and sequential standard static space-times.

Gradient Ricci-harmonic solitons on doubly warped product manifolds

We give necessary and sufficient conditions for doubly warped product manifolds to be gradient Ricci-harmonic solitons. We also give a physical application for this kind of solitons.

On the geometry of Einstein-type structures

The aim of the paper is to study the geometry of a Riemannian manifold M, with a special structure depending on 3 real parameters, a smooth map φ into a target Riemannian manifold N, and a smooth function f on M itself. We will occasionally let some of the parameters be smooth functions. For a special value of one of them, the structure is obtained by a conformal deformation of a harmonic-Einstein manifold. The setting generalizes various previously studied situations; for instance, Ricci solitons, Ricci harmonic solitons, generalized quasi-Einstein manifolds and so on. One main ingredient of our analysis is the study of certain modified curvature tensors on M, related to the map φ, and to develop a series of results for harmonic-Einstein manifolds that parallel those obtained for Einstein manifolds both some time ago and in the very recent literature. We then turn to locally characterize, via a couple of integrability conditions and mild assumptions on f, the manifold M as a warped product with harmonic-Einstein fibers extending in a very non trivial way a recent result for Ricci solitons. We then consider rigidity and non existence, both in the compact and non-compact cases. This is done via integral formulas and, in the non-compact case, via analytical tools previously introduced by the authors.

On the spectrum of the weighted p-Laplacian under the Ricci-harmonic flow

This paper studies the behaviour of the spectrum of the weighted p-Laplacian on a complete Riemannian manifold evolving by the Ricci-harmonic flow. Precisely, the first eigenvalue diverges in a finite time along this flow. It is further shown that the same divergence result holds on gradient shrinking and steady almost Ricci-harmonic solitons under the condition that the soliton function is nonnegative and superharmonic. We also continue the program in (Abolarinwa, Adebimpe and Bakare in J. Ineq. Appl. 2019:10, 2019) to the case of volume-preserving Ricci-harmonic flow.

On the geometry of φ-curvatures

The aim of this paper is to continue the geometric-analytic study of φ-curvatures initiated in [2]. These curvatures arise naturally in many geometric contexts, notably in Ricci-harmonic solitons theory. In the present paper we prove two rigidity results related to harmonic-Einstein manifolds, a generalization of the notion of Einstein manifolds to the present situation. We observe that, when we restrict our theorems to the classical case of Einstein manifolds, we obtain some new results even in this setting.

Gap theorems for compact almost Ricci-harmonic solitons

Almost Ricci-harmonic solitons are generalization of Ricci-harmonic solitons, almost Ricci solitons and harmonic-Einstein metrics. The main focus of this paper is to establish necessary and sufficient conditions for a gradient shrinking almost Ricci-harmonic soliton on a compact domain to be almost harmonic-Einstein.

On the Entropy Formulas and Solitons for the Ricci-Harmonic Flow

This paper is concerned with entropy monotonicity formulas and classification of gradient almost soliton for Ricci-harmonic flow. The new entropy formulas introduced here are monotone in general along the flow and constant exactly on shrinker or expander as the case may be. The consequence of this is the nonexistence of periodic solutions except those that are gradient solitons. Furthermore, the paper discusses gradient almost Ricci-harmonic soliton with respect to a fixed metric, not minding the dynamical nature of the flow but treating the defining elliptic equations and relying on analytic techniques. Finally, we are able to classify gradient almost Ricci-harmonic solitons.

Scalar curvature bound and compactness results for Ricci harmonic solitons

Scalar curvature bound and compactness results for Ricci harmonic solitons

Volume Growth of Shrinking Gradient Ricci-Harmonic Soliton

In this paper, we study the shrinking gradient Ricci-harmonic soliton. Firstly using Chow–Lu–Yang’s argument, we give a necessary and sufficient condition for complete noncompact shrinking gradient Ricci-harmonic solitons with $$S\ge \delta $$ to have polynomial volume growth with order $$n-2\delta $$ . Secondly, we derive a Logarithmic Sobolev inequality, as an application, we prove that any noncompact shrinking gradient Ricci-harmonic soliton must have linear volume growth, generalizing previous result of Munteanu and Wang (Commun Anal Geom 20(1):55–94, 2012).

On the relation between Ricci-Harmonic solitons and Ricci solitons

Let (Mm,gij) and (Nn,hβγ) be two Riemannian manifolds, and ϕ:M→N a smooth map. By definition, a gradient Ricci-Harmonic soliton satisfies(0.1){Rij−α∇iϕ∇jϕ+∇i∇jf=λgij;τgϕ=∇iϕ∇if, for some f∈C∞(M) and constants α and λ. Here τgϕ=trg(∇dϕ) is the tension filed of ϕ. We prove that when α>0 and the sectional curvature of N is bounded from above by αm, any shrinking or steady Ricci-Harmonic soliton (i.e., λ>0 or λ=0, respectively) must be a Ricci soliton, namely, ϕ is a constant map. In particular, it implies that the shrinking and steady solitons generated from Bernhard List's flow [9] are exactly the corresponding solitons of the Ricci flow, and hence some recent results regarding the shrinking solitons of List's flow are actually duplications of the previous results for Ricci solitons.

On f-non-parabolic ends for Ricci-harmonic metrics

In this paper, we study gradient Ricci-harmonic soliton metrics and quasi Ricci-harmonic metrics (both metrics are called Ricci-harmonic). First, we prove that all ends of \(\tau \)-quasi Ricci-harmonic metrics with \(\tau >1\) should be f-non-parabolic if \(\lambda =0,\mu >0\), or \(\lambda <0, \mu \ge 0\). For the case that \(\lambda<0, \mu < 0\), we can also arrive at the f-non-parabolic property if we add a condition about the scalar curvature. Furthermore, we discuss the connectivity at infinity for quasi Ricci-harmonic metrics. We also conclude that all ends of steady or expanding gradient Ricci-harmonic solitons should be f-non-parabolic, based on which we establish structure theorems for these two solitons.

Gap theorems for Ricci-harmonic solitons

In this paper, using estimates for the generalized Ricci curvature, we shall give some gap theorems for Ricci-harmonic solitons with compact domain manifolds by showing some necessary and sufficient conditions for the solitons to be harmonic-Einstein. Our results may be regarded as generalizations of recent works by H. Li, and M. Fernandez-Lopez and E. Garcia-Rio.

A lower diameter bound for compact domain manifolds of shrinking Ricci-harmonic solitons

In this paper, we shall give a lower diameter bound for compact domain manifolds of shrinking Ricci-harmonic solitons. Our result may be regarded as a generalization to Ricci-harmonic geometry of the recent works by Fernández-López and García-Río (Q. J. Math. 61, 319-327, 2010), Futaki and Sano (Asian J. Math. 17, 17-32, 2013), and Futaki et al. (Ann. Global Anal. Geom. 44, 105-114, 2013).

Volume growth for gradient shrinking solitons of Ricci-harmonic flow

In this paper, we derive an estimate on the potential functions of complete noncompact gradient shrinking solitons of Ricci-harmonic flow, and show that complete noncompact gradient shrinking Ricci-harmonic solitons have Euclidean volume growth at most.