相关论文: On the classification of four-dimensional gradient…

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…

In this article, we study four-dimensional complete gradient shrinking Ricci solitons. We prove that a four-dimensional complete gradient shrinking Ricci soliton satisfying a pointwise condition involving either the self-dual or…

We prove that a gradient shrinking Ricci soliton with fourth order divergence-free Riemannian tensor is rigid. For the $4$-dimensional case, we show that any gradient shrinking Ricci soliton with fourth order divergence-free Riemannian…

Let $(M, g, f)$ be a $4$-dimensional complete noncompact gradient shrinking Ricci soliton with the equation $Ric+\nabla^2f=\lambda g$, where $\lambda$ is a positive real number. We prove that if $M$ has constant scalar curvature…

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 show that a four-dimensional complete gradient shrinking Ricci soliton with positive isotropic curvature is either a quotient of S^4 or a quotient of S^3 cross R. This gives a clean classification result removing the earlier additional…

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…

In this note we prove that any four-dimensional half conformally flat gradient steady Ricci soliton must be either Bryant's soliton or Ricci flat. We also classify four-dimensional half conformally flat gradient shrinking 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…

This is a sequel to our paper [24], in which we investigated the geometry of 4-dimensional gradient shrinking Ricci solitons with half positive (nonnegative) isotropic curvature. In this paper, we mainly focus on 4-dimensional gradient…

In this note, we study the classification of four-dimensional complete gradient steady and expanding Ricci solitons. Specifically, under the asymptotically cylindrical (respectively, asymptotically conical) assumption, we classify gradient…

In this paper, we have proved that if a complete conformally flat gradient shrinking Ricci soliton has linear volume growth or the scalar curvature is finitely integrable and also the reciprocal of the potential function is subharmonic,…

Suppose $(M, g, f)$ is a 5-dimensional complete shrinking gradient Ricci soliton with $R=1$. If it has bounded curvature, we prove that it is a finite quotient of $\mathbb{R}^3\times \mathbb{S}^2$.

Haslhofer and M\"uller proved a compactness Theorem for four-dimensional shrinking gradient Ricci solitons, with the only assumption being that the entropy is uniformly bounded from below. However, the limit in their result could possibly…

In this survey paper, we analyse and compare the recent curvature estimates for three types of $4$-dimensional gradient Ricci solitons, especially between Ricci shrinkers [58] and expanders [17]. In addition, we provide some new curvature…

In this paper, we investigate the geometry of 4-dimensional complete gradient shrinking Ricci solitons with half positive isotropic curvature (half PIC) or half nonnegative isotropic curvature. Our first main result is a certain form of…

As of today, there are very few known complete shrinking Ricci solitons in dimension 4, and all examples discovered so far are K\"ahler and/or Einstein. In this note, we prove that any four dimensional J-invariant gradient shrinking Ricci…

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 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…

In this article, we investigate four-dimensional gradient shrinking Ricci solitons close to a K\"ahler model. The first theorem could be considered as a rigidity result for the K\"ahler-Ricci soliton structure on $\mathbb{S}^2\times…