Related papers: Convergence of compact Ricci solitons
A closed four dimensional manifold cannot possess a non-flat Ricci soliton metric with arbitrarily small $L^2$-norm of the curvature. In this paper, we localize this fact in the case of shrinking Ricci solitons by proving an…
We estimate from above the rate at which a solution to the normalized Ricci flow on a closed manifold may converge to a limit soliton. Our main result implies that any solution which converges modulo diffeomorphisms to a soliton faster than…
We show that for an $n$ dimensional complete non Ricci flat gradient steady Ricci soliton with potential function $f$ bounded above by a constant and curvature tensor $Rm$ satisfying $\overline{\lim}_{r\to \infty} r|Rm|<\frac{1}{5}$, then…
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,…
In this survey article, we discuss some topics on self-similar solutions to the Ricci flow and the mean curvature flow. Self-similar solutions to the Ricci flow are known as Ricci solitons. In the first part of this paper we discuss a lower…
We first investigate the asymptotics of conical expanding gradient Ricci solitons by proving sharp decay rates to the asymptotic cone both in the generic and the asymptotically Ricci flat case. We then establish a compactness theorem…
A question about Ricci flow is when the diameters of the manifold under the evolving metrics stay finite and bounded away from 0. Topping \cite{T:1} addresses the question with an upper bound that depends on the $L^{(n-1)/2}$ bound of the…
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…
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…
Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete, conformally compact metrics. Related to this issue, an…
We derive new, sharp lower bounds for certain curvature functionals on the space of Riemannian metrics of a smooth compact 4-manifold with a non-trivial Seiberg-Witten invariant. These allow one, for example, to exactly compute the infimum…
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…
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 paper, we extend the theory of Ricci flows satisfying a Type-I scalar curvature condition at a finite-time singularity. In [Bam16], Bamler showed that a Type-I rescaling procedure will produce a singular shrinking gradient Ricci…
We introduce new families of four-dimensional Ricci solitons of cohomogeneity two with volume collapsing ends. In a local presentation of the metric conformal to a product, we reduce the soliton equation to a degenerate Monge-Amp\`{e}re…
In this paper, we study the following conjecture of Hamilton: Any compact gradient shrinking Ricci soliton with positive curvature operator must be Einstein. We first derive several identities. Then we show that the conjecture is true under…
In this paper, we give a description for steady Ricci solitons with a linear decay of sectional curvature. In particular, we classify all 3-dimensional steady Ricci solitons and 4-dimensional $\kappa$-noncollpased steady Ricci solitons with…
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 and expose all the gradient Ricci solitons on complete surfaces, open or closed, with curvature bounded below, and possibly with a discrete set of cone-like singular points that arise naturally. We give a precise qualitative…
In this paper, we study the rigidity of eigenvalues of shring Ricci solitons. It is known that the drifted Laplacian on shrinking Ricci solitons has discrete spectrum, its eigenvalues have a lower bound and a rigidity result holds. Firstly,…