Related papers: Hamilton-Ivey estimates for gradient Ricci soliton…

We describe three-dimensional Lorentzian homogeneous Ricci solitons, showing that all types (i.e. shrinking, expanding and steady) exist. Moreover, all non-trivial examples have non-diagonalizable Ricci operator with one only eigenvalue.

In this paper, we study the fundamental group of the complete steady gradient Ricci soliton with nonnegative sectional curvature. We prove that the fundamental group of such a Ricci soliton is either trivial or infinite. As a corollary, we…

We prove that any noncompact $\kappa$-noncollapsed steady Ricci soliton with nonnegative curvature operator must be rotationally symmetric if it has a linear curvature decay.

Let $(M^4, g, f)$ be a four-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+\nabla^2f= \frac{1}{2}g$. If its scalar curvature is $1$, Cheng-Zhou \cite{Cheng-Zhou} proved that it is a finite quotient…

We study stability of non-compact gradient Kaehler-Ricci flow solitons with positive holomorphic bisectional curvature. Our main result is that any compactly supported perturbation and appropriately decaying perturbations of the Kaehler…

In this paper we take a look at conditions that make a Riemann soliton trivial, compacity being one of them. We also show that the behaviour at infinity of the gradient field of a non-compact gradient Riemann soliton might cause the soliton…

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,…

The aim of this paper is to prove some classification results for generic shrinking Ricci solitons. In particular, we show that every three dimensional generic shrinking Ricci soliton is given by quotients of either $\mathds{S}^3$,…

In this paper we consider connections between Ricci solitons and Einstein metrics on homogeneous spaces. We show that a semi-algebraic Ricci soliton admits an Einstein one-dimensional extension if the soliton derivation can be chosen to be…

We prove that the four-dimensional blowdown shrinking Ricci soliton constructed by Feldman-Ilmanen-Knopf is strictly linearly stable in the sense of Cao-Hamilton-Ilmanen. This provides the first known example of a non-cylindrical linearly…

In the paper, we analysis the asymptotic behavior of noncompact $\kappa$-noncollapsed steady gradient Ricci soliton $(M, g)$ with nonnegative curvature operator away from a compact set $K$ of $M$. In particular, we prove: any $4d$…

We prove that any gradient shrinking Ricci soliton has at most Euclidean volume growth. This improves a recent result of H.-D. Cao and D. Zhou by removing a condition on the growth of scalar curvature.

In this paper, we shall give a new upper diameter estimate for complete Riemannian manifolds in the case that the Bakry-\'Emery Ricci curvature has a positive lower bound and the norm of the potential function has an upper bound. Our…

Gradient almost para-Ricci-like solitons on para-Sasaki-like Riemannian $\Pi$-manifolds are studied. It is proved that these objects have constant soliton coefficients. For the soliton under study is shown that the corresponding scalar…

In this short note, using G\"unther's volume comparison theorem and Yokota's gap theorem on complete shrinking gradient Ricci solitons, we prove that for any complete shrinking gradient Ricci soliton $(M^{n},g,f)$ with sectional curvature…

In this paper, I computed the second variation formula of the generalized Einstein-Hilbert functional and prove that a Bismut-flat, Einstein manifold is linearly stable under some curvature assumption. In the last part of the paper, I prove…

In this work, we construct several sequences of metrics on sphere with different limiting behaviors. By combining with the work of Deruelle, we use it and the localized maximum principle to construct various examples of expanding gradient…

In this paper, we prove some rigidity theorems for shrinking gradient Ricci solitons with nonnegative sectional curvature.

We show that if two gradient Ricci solitons are asymptotic along some end of each to the same regular cone, then the soliton metrics must be isometric on some neighborhoods of infinity of these ends. Our theorem imposes no restrictions on…

It is shown that locally conformally flat Lorentzian gradient Ricci solitons are locally isometric to a Robertson-Walker warped product, if the gradient of the potential function is non null, and to a plane wave, if the gradient of the…