Related papers: Two-dimensional gradient Ricci solitons revisited
We show that a shrinking Ricci soliton with positive sectional curvature must be compact. This extends a result of Perelman in dimension three and improves a result of Naber in dimension four, respectively.
We develop a method for constructing complete gradient Ricci solitons realized as fiber bundles endowed with warped metrics, and we establish necessary and sufficient conditions for their existence. As an application, we present new…
In Riemannian geometry, Ricci soliton inequalities are an important field of study that provide profound insights into the geometric and analytic characteristics of Riemannian manifolds. An extensive study of Ricci soliton inequalities is…
In the paper, we study complete almost Ricci solitons using the concepts and methods of geometric dynamics and geometric analysis. In particular, we characterize Einstein manifolds in the class of complete almost Ricci solitons. Then, we…
In this paper we prove that a nonflat K\"{a}hler-Ricci soliton of the Ricci flow on a complex two-dimensional K\"{a}hler manifold with nonnegative holomorphic bisectional curvature can not be of maximal volume growth.
The purpose of this article is to study an $f$-cosymplectic manifold $M$ admitting Ricci solitons. Here we consider mainly two classes of Ricci solitons on $f$-cosymplectic manifolds. One is the class of contact Ricci solitons. The other is…
In this paper, we study gradient Ricci expanding solitons $(X,g)$ satisfying $$ Rc=cg+D^2f, $$ where $Rc$ is the Ricci curvature, $c<0$ is a constant, and $D^2f$ is the Hessian of the potential function $f$ on $X$. We show that for a…
Ricci-like solitons with arbitrary potential are introduced and studied on Sasaki-like almost contact B-metric manifolds. It is proved that the Ricci tensor of such a soliton is the vertical component of both B-metrics multiplied by a…
In this paper we give some results on the topology of manifolds with $\infty$-Bakry-\'Emery Ricci tensor bounded below, and in particular of steady and expanding gradient Ricci solitons. To this aim we clarify and further develop the theory…
We investigate four-dimensional gradient shrinking Ricci solitons with positive modified sectional curvature. Our first main result shows that if the norm of the self-dual Weyl tensor and the scalar curvature satisfy a certain sharp…
In this article, we investigate the geometry of $4$-dimensional compact gradient Ricci solitons. We prove that, under an upper bound condition on the range of the potential function, a $4$-dimensional compact gradient Ricci soliton must…
We study three-dimensional generalized Ricci solitons, both in Riemannian and Lorentzian settings. We shall determine their homogeneous models, classifying left-invariant generalized Ricci solitons on three-dimensional Lie groups.
The Eisenhart problem of finding parallel tensors treated already in the framework of quasi-constant curvature manifolds in \cite{x:j} is reconsidered for the symmetric case and the result is interpreted in terms of Ricci solitons. If the…
In this paper we consider $4$-dimensional steady soliton singularity models, i.e., complete steady gradient Ricci solitons that arise as the rescaled limit of a finite time singular solution of the Ricci flow on a closed $4$-manifold. In…
We consider almost Riemann and almost Ricci solitons in a $D$-homothetically deformed Kenmotsu manifold having as potential vector field a gradient vector field, a solenoidal vector field or the Reeb vector field of the deformed structure,…
Using center manifolds and topological degree theory, we construct a new family of complete, $SU(2)$-invariant and steady gradient Ricci solitons on the four-dimensional non-compact cohomogeneity one manifold with group diagram…
Any homogeneous expanding Ricci soliton is known to be isometric to a Lie subgroup of the solvable part of the Iwasawa decomposition associated with a symmetric space of non-compact type, with the metric induced as a submanifold. In this…
We study both function theoretic and spectral properties of the weighted Laplacian $\Delta_f$ on complete smooth metric measure space $(M,g,e^{-f}dv)$ with its Bakry-\'{E}mery curvature $Ric_f$ bounded from below by a constant. In…
For all complex dimensions n>=2, we construct complete Kaehler manifolds of bounded curvature and non-negative Ricci curvature whose Kaehler--Ricci evolutions immediately acquire Ricci curvature of mixed sign.
We describe four-dimensional Lorentzian algebraic Ricci solitons. In sharp contrast with the Riemannian situation, any connected and simply connected four-dimensional Lie group admits a left-invariant Lorentz metric which is a Ricci…