Related papers: On Four-dimensional Steady gradient Ricci solitons…

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…

We consider gradient Ricci solitons conformal to a $n$-dimensional pseudo-Euclidean space and we completely describe the most general ansatz that reduces the resulting system of partial differential equations to a system of ordinary…

[Dedicated to Richard S. Hamilton on forty years of Ricci flow] Gradient Ricci solitons have garnered significant attention both as self-similar solutions and singularity models of the Ricci flow. This survey article starts with a list of…

In this paper we study the gradient Ricci shrinking soliton equation on rotationally symmetric manifolds of dimension three and higher and prove that the only complete examples of such metrics on $S^n$, $\R{n}$ and $\R{}\times S^{n-1}$ are,…

We construct a family of non-collapsed, non-K\"ahler, non-Einstein steady Ricci solitons in even dimensions greater or equal to four. These solitons exist on complex line bundles over K\"ahler-Einstein manifolds of positive scalar…

We prove that a four-dimensional gradient shrinking Ricci soliton with $\delta W^{\pm}=0$ is either Einstein, or a finite quotient of $S^3\times\mathbb{R}$, $S^2\times\mathbb{R}^2$ or $\mathbb{R}^4$. We also prove that a four-dimensional…

We establish a dichotomy on the curvature decay for four dimensional complete noncompact non Ricci flat steady gradient Ricci soliton with linear curvature decay and proper potential function. A similar dichotomy is also shown in higher…

In this paper, we classify n-dimensional (n>3) complete Bach-flat gradient shrinking Ricci solitons. More precisely, we prove that any 4-dimensional Bach-flat gradient shrinking Ricci soliton is either Einstein, or locally conformally flat…

We study the stability of non compact steady and expanding gradient Ricci solitons. We first show that strict linear stability implies dynamical stability. Then we give various sufficient geometric conditions ensuring the strict linear…

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…

The aim of this paper is to study geometrical aspects of static spacetime admitting an almost gradient Ricci soliton. Among others, We first determine the conditions under which the base manifold of static spacetime possess an almost…

The object of the present paper is to study some types of Ricci pseudosymmetric $(LCS)_n$-manifolds whose metric is Ricci soliton. We found the conditions when Ricci soliton on concircular Ricci pseudosymmetric, projective Ricci…

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…

In this paper we prove new classification results for nonnegatively curved gradient expanding and steady Ricci solitons in dimension three and above, under suitable integral assumptions on the scalar curvature of the underlying Riemannian…

For any 3D steady gradient Ricci soliton with positive curvature, we prove that it must be isometric to the Bryant soliton if it is asymptotic to a ray. Otherwise, it is asymptotic to a sector and hence a flying wing. We show that all 3D…

Our main aim in this paper is to investigate the rigidity of complete noncompact gradient steady Ricci solitons with harmonic Weyl tensor. More precisely, we prove that an $n$-dimensional ($n\geq 5$) complete noncompact gradient steady…

We classify complete gradient Ricci solitons satisfying a fourth-order vanishing condition on the Weyl tensor, improving previously known results. More precisely, we show that any $n$-dimensional ($n\geq 4$) gradient shrinking Ricci soliton…

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…

In this paper, we prove the compactness theorem for gradient Ricci solitons. Let $(M_{\alpha}, g_{\alpha})$ be a sequence of compact gradient Ricci solitons of dimension $n\geq 4$, whose curvatures have uniformly bounded $L^{\frac{n}{2}}$…

We describe the structure of the Ricci tensor on a locally homogeneous Lorentzian gradient Ricci soliton. In the non-steady case, we show the soliton is rigid in dimensions three and four. In the steady case, we give a complete…