Related papers: On normal forms of gradient Ricci 4-solitons

Some observations about the local and global generality of gradient Kahler Ricci solitons are made, including the existence of a canonically associated holomorphic volume form and vector field, the local generality of solutions with a…

We first show that a K\"ahler cone appears as the tangent cone of a complete expanding gradient K\"ahler-Ricci soliton with quadratic curvature decay with derivatives if and only if it has a smooth canonical model (on which the soliton…

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…

On an oriented 4-manifold, we examine the geometry that arises when the curvature operator of a Riemannian or Lorentzian metric $g$ commutes, not with its own Hodge star operator, but rather with that of another semi-Riemannian metric $h$…

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 $1996$, H.-D. Cao constructed a $U(n)$-invariant steady gradient K\"ahler-Ricci soliton on $\mathbb{C}^{n}$ and asked whether every steady gradient K\"ahler-Ricci soliton of positive curvature on $\mathbb{C}^{n}$ is necessarily…

A new approach to normal operators in real Hilbert spaces is discussed, and a spectral representation is obtained, derived directly from the complex case. The results are then applied to quaternionic normal operators, regarded as a special…

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…

This paper is concerned with the study of generalized gradient Ricci-Yamabe solitons. We characterize the compact generalized gradient Ricci-Yamabe soliton and find certain conditions under which the scalar curvature becomes constant. The…

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…

We show that the generalized K\"ahler-Ricci soliton equation on 4-dimensional toric K\"ahler orbifolds reduces to ODEs assuming there is a Hamiltonian 2-form. This leads to an explicit resolution of this equation on labeled triangles and…

We consider four dimensional conformally flat homogeneous pseudo Riemannian manifolds. According to forms (Seger types) of the Ricci operator, we provide a full classification of four dimensional pseudo Riemannian conformally flat…

Let $H$ be a complex separable Hilbert space and $B(H)$ the algebra of all bounded linear operators on $H$. In this paper, we give considerable generalizations of the inequalities for norms of commutators of normal operators. Let $S, T \in…

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 give a delay estimate of scalar curvature for a complete non-compact expanding (or steady) gradient Ricci soliton with nonnegative Ricci curvature. As an application, we prove that any complete non-compact expanding (or…

We show that, up to the flow of the soliton vector field, there exists a unique complete steady gradient K\"ahler-Ricci soliton in every K\"ahler class of an equivariant crepant resolution of a Calabi-Yau cone converging at a polynomial…

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…

We study the non Ricci flat gradient steady K\"{a}hler Ricci soliton with non-negative Ricci curvature and weak integrability condition of the scalar curvature $S$, namely $\underline{\lim}_{r\to \infty} r^{-1}\int_{B_r} S=0$, and show that…

Consider the standard action of $SO(2)\times SO(3)$ on $\mathbb{R}^5=\mathbb{R}^2\oplus \mathbb{R}^3$. We establish the existence of a uniform constant $\mathcal{C}>0$ so that any $SO(2)\times SO(3)$-invariant Ricci soliton on…

The purpose of the article is to investigate the existence of Ricci solitons and the nature of curvature inheritance as well as collineations on the Robinson-Trautman (briefly, RT) spacetime. It is shown that under certain conditions RT…