Related papers: On kappa-noncollapsed complete noncompact shrinkin…
In this paper we show how techniques coming from stochastic analysis, such as stochastic completeness (in the form of the weak maximum principle at infinity), parabolicity and $L^p$-Liouville type results for the weighted Laplacian…
We show that Lorentzian manifolds whose isometry group is of dimension at least $\frac{1}{2}n(n-1)+1$ are expanding, steady and shrinking Ricci solitons and steady gradient Ricci solitons. This provides examples of complete locally…
In this paper, we prove that a gradient shrinking compact K\"ahler-Ricci soliton cannot have too large Ricci curvature unless it is K\"ahler-Einstein.
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 show that recent work of Ni and Wilking yields the result that a noncompact nonflat Ricci shrinker has at most quadratic scalar curvature decay. The examples of noncompact K\"{a}hler--Ricci shrinkers by Feldman, Ilmanen, and Knopf…
In this paper we study the behavior of the scalar curvature at infinity on complete noncompact steady gradient Ricci solitons. In dimension four, we assume that the canonical Ricci flow induced by the soliton is a weak $\kappa$-solution and…
In this paper, an n-dimensional complete open manifold with nonnegative Ricci curvature and collapsing volume has been investigated. If its radial sectional curvature bounded from below, it shows that such a manifold is of finite…
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…
We study $3$-dimensional Ricci solitons which project via a semi-conformal mapping to a surface. We reformulate the equations in terms of parameters of the map; this enables us to give an ansatz for constructing solitons in terms of data on…
Let $(M^n, g, f)$, $n\geq 5$, be a complete gradient expanding Ricci soliton with nonnegative Ricci curvature $Rc\geq 0$. In this paper, we show that if the asymptotic scalar curvature ratio of $(M^n, g, f)$ is finite (i.e., $ \limsup_{r\to…
Geometrical aspects of a perfect fluid spacetime are described in terms of different curvature tensors and $\eta$-Ricci and $\eta$-Einstein solitons in a perfect fluid spacetime are determined. Conditions for the Ricci soliton to be steady,…
A lower-bound estimate of injectivity radius for complete Riemannian manifolds is discussed in a pure geometric viewpoint and is applied to study tangent cones at infinity of certain gradient Ricci solitons. We also study the asymptotic…
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…
This paper studies gradient almost Ricci-harmonic soliton with respect to a fixed metric. We rely on analytic techniques to estabilish some basic elliptic and integral equations for the structure of almost Ricci-harmonic soliton which…
Regarding Ricci flow as a dynamical system, we derive sufficient conditions for noncompact stationary (Ricci-flat) solutions to possess infinite-dimensional unstable manifolds, and provide examples satisfying those criteria that have…
In this paper, we study the topology of complete noncompact Riemannian manifolds with asymptotically nonnegative Ricci curvature and large volume growth. We prove that they have finite topological types under some curvature decay and volume…
In this note we show that, under certain curvature positivity conditions (the weak $\operatorname{PIC}-2$ condition or the nonnegative bisectional curvature condition), a complete and noncompact expanding breather of the Ricci flow is also…
In this paper, we establish a compactness theorem for gradient Ricci solitons with scalar curvature bounds and uniform lower bounds of harmonic coordinates. Our approach is to bootstrap regularity in harmonic coordinates by exploiting the…
In this paper, we prove that any compact 2-sided smooth stable minimal hypersurface in gradient Ricci soliton $(M^{n},g,f)$ with scalar curvature $R\geq(n-1)\lambda$ must have vanished second fundamental form and vanished normal Ricci…
We show that for a complete Ricci shrinker there exists a sequence of points tending to infinity whose norms of the Ricci tensor grow at most linearly.