Related papers: Geometry of Complete Gradient Shrinking Ricci Soli…

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…

Some aspects of the multidimensional soliton geometry are considered.

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…

We carry out a Painlev\'e analysis of the systems of differential equations corresponding to the steady and the expanding, rotationally symmetric, gradient Ricci solitons on $\mathbb{R}^n$. For the steady case, dimensions of the form…

We show that a shrinking Ricci soliton with positive sectional curvature must be compact. This extends a result of Perelman in dimension three and improves a result of Naber in dimension four, respectively.

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 rigorously analyze the scalar curvature of complete expanding gradient Yamabe solitons. We completely classify nontrivial complete expanding gradient Yamabe solitons in both cases: when the scalar curvature is greater than…

Ricci-like solitons with arbitrary potential are introduced and studied on Sasaki-like almost contact B-metric manifolds. It is proved that the Ricci tensor of such a soliton is the vertical component of both B-metrics multiplied by a…

We summarize the fall-off of electromagnetic and gravitational fields in n>5 dimensional Ricci-flat spacetimes along an asympotically expanding non-singular geodesic null congruence.

In this note, we obtain a sharp volume estimate for complete gradient Ricci solitons with scalar curvature bounded below by a positive constant. Using Chen-Yokota's argument we obtain a local lower bound estimate of the scalar curvature for…

We will show that if a gradient shrinking Ricci soliton has an approximate symmetry on one scale, this symmetry propagates to larger scales. This is an example of the shrinker principle which roughly states that information radiates…

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…

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…

In this article we study the limiting behavior of the K\"ahler Ricci flow on complete non-compact K\"ahler manifolds. We provide sufficient conditions under which a complete non-compact gradient K\"ahler-Ricci soliton is biholomorphic to…

In this paper, we study gradient Ricci expanding solitons $(X,g)$ satisfying $$ Rc=cg+D^2f, $$ where $Rc$ is the Ricci curvature, $c<0$ is a constant, and $D^2f$ is the Hessian of the potential function $f$ on $X$. We show that for a…

We consider three- and four-dimensional pseudo-Riemannian generalized symmetric spaces, whose invariant metrics were explicitly described in [15]. While four-dimensional pseudo-Riemannian generalized symmetric spaces of types A, C and D are…

The present paper is devoted to the study a global aspect of the geometry of harmonic mappings and, in particular, infinitesimal harmonic transformations, and represents the application of our results to the theory of Ricci solutions and…

We prove that a shrinking gradient Ricci soliton which is asymptotic to a K\"ahler cone along some end is itself K\"ahler on some neighborhood of infinity of that end. When the shrinker is complete, it is globally K\"ahler.

In this article, we study properly immersed complete noncompact submanifolds in a complete shrinking gradient Ricci soliton with weighted mean curvature vector bounded in norm. We prove that such a submanifold must have polynomial volume…

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…