相关论文: On rotationally invariant shrinking gradient Ricci…

In this paper, we study the gradient Ricci soliton equation on a complete Riemannian manifold. We show that under a natural decay condition on the Ricci curvature, the Ricci soliton is Ricci-flat and ALE.

In this paper, we will study the asymptotic geometry of 4-dimensional steady gradient Ricci solitons under the condition that they dimension reduce to $3$-manifolds. We will show that such 4-dimensional steady gradient Ricci solitons either…

Let (M,g) be a steady gradient Ricci soliton of dimension n \geq 4 which has positive sectional curvature and is asymptotically cylindrical. Under these assumptions, we show that (M,g) is rotationally symmetric. In particular, our result…

We give a new proof for the existence of rotationally symmetric steady and expanding gradient Ricci solitons in dimension $n+1$, $2\le n\le 4$, with metric $g=\frac{da^2}{h(a^2)}+a^2d\,\sigma$ for some function $h$ where $d\sigma$ is the…

In this paper, we study gradient Ricci soitons on smooth orbifolds. We prove that the scalar curvature of a complete shrinking or steady gradient Ricci soliton on an orbifold is nonnegative. We also show that a complete…

We study the modified Ricci solitons as a new class of Einstein type metrics that contains both Ricci solitons and $n$-quasi-Einstein metrics. This class is closely related to the construction of the Ricci solitons that are realised as…

In this paper, we survey the volume growth estimates for shrinking, steady, and expanding gradient Ricci solitons. Together with the known results, we also prove some new volume growth estimates for expanding gradient Ricci solitons.

In this paper we classify the four dimensional gradient shrinking solitons under certain curvature conditions satisfied by all solitons arising from finite time singularities of Ricci flow on compact four manifolds with positive isotropic…

We study gradient Ricci solitons with maximal symmetry. First we show that there are no non-trivial homogeneous gradient Ricci solitons. Thus the most symmetry one can expect is an isometric cohomogeneity one group action. Many examples of…

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 $L^{\frac{n}{2}}$…

The main purpose of this paper is to investigate the curvature behavior of four dimensional shrinking gradient Ricci solitons. For such soliton $M$ with bounded scalar curvature $S$, it is shown that the curvature operator $\mathrm{Rm}$ of…

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…

All known examples of nontrivial homogeneous Ricci solitons are left-invariant metrics on simply connected solvable Lie groups whose Ricci operator is a multiple of the identity modulo derivations (called solsolitons, and nilsolitons in the…

In this paper, we prove some classification results for four-dimensional gradient Ricci solitons. For a four-dimensional gradient shrinking Ricci soliton with $div^4Rm^\pm=0$, we show that it is either Einstein or a finite quotient of…

We study deformations of shrinking Ricci solitons on a compact manifold M, generalising the classical theory of deformations of Einstein metrics. Using appropriate notions of twisted slices S_f inside the space of all Riemannian metrics on…

Let $(Y,d)$ be a Gromov-Hausdorff limit of closed shrinking Ricci solitons with uniformly upper bounded diameter and lower bounded volume. We prove that off a closed subset of codimension at least 2, $Y$ is a smooth manifold satisfying a…

In this paper, we study solitons on $3$-dimensional manifolds. In particular, we show that $3$-dimensional pseudo-symmetric gradient Ricci solitons and nontrivial gradient Yamabe solitons are locally isometric to either $\mathbb{R}^{3}$,…

In this paper, we study conformal Ricci solitons and conformal gradient Ricci solitons on generalized ($\kappa,\mu$)-space forms. The conditions for the solitons to be shrinking, steady, and expanding are derived in terms of conformal…

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…

In this work, we study metrics which are both homogeneous and Ricci soliton. If there exists a transitive solvable group of isometries on a Ricci soliton, we show that it is isometric to a solvsoliton. Moreover, unless the manifold is flat,…