Related papers: Splitting theorem of Gradient $\rho$-Einstein soli…

We develop heat kernel and Green's function estimates for manifolds with positive bottom spectrum. The results are then used to establish existence and sharp estimates of the solution to the Poisson equation on such manifolds with Ricci…

In this paper we investigate gradient Yamabe solitons, either steady or shrinking, that can be isometrically immersed into space forms as hypersurfaces that admit an upper bound on the norm of their second fundamental form. Those solitons…

We construct exhaustion and cut-off functions with controlled gradient and Laplacian on manifolds with Ricci curvature bounded from below by a (possibly unbounded) nonpositive function of the distance from a fixed reference point, without…

We characterize complete nonnegatively curved steady gradient soliton with curvature in L^1. We show that there are isometric to a product (R^2,g_{cigar}) times(R^{n-2}, eucl))/Gamma where Gamma is a Bieberbach group of rank n-2. We prove…

In this paper, we explore the limit structure of a sequence of Riemannian manifolds with Bakry-\'Emery Ricci curvature bounded below in the Gromov-Hausdorff topology. By extending the techniques established by Cheeger-Cloding for Riemannian…

We first show that any $4$-dimensional non-Ricci-flat steady gradient Ricci soliton singularity model must satisfy $|Rm|\leq cR$ for some positive constant $c$. Then, we apply the Hamilton-Ivey estimate to prove a quantitative lower bound…

We establish variational formulas for Ricci upper and lower bounds, as well as a derivative formula for the Ricci curvature. As applications, constant curvature manifolds, Einstein manifolds and Ricci parallel manifolds are identified,…

Suppose $(M^n, g, f)$ is a complete shrinking gradient Ricci soliton. We give several rigidity results under some natural conditions, generalizing the results in \cite{Petersen-Wylie,Guan-Lu-Xu}. Using maximum principle, we prove that…

In this paper we prove an area comparison result for certain totally geodesic surfaces in 3-manifolds with a lower bound on the scalar curvature. This result is a variant of a comparison theorem of Heintze-Karcher for minimal hypersurfaces…

We give a result estimating the dimension of the Lie algebra of Killing vector fields on an irreducible non-trivial gradient Ricci soliton. Then we study the structure of this manifold when the maximal dimension is attained. There are local…

In this paper we prove an upper bound for the bottom of the spectrum of the Laplacian on manifolds with Ricci curvature bounded in integral sense. Our arguments rely on the existence of a minimal positive Green's function and its…

In this paper we show that a $\rho$-Einstein solitons conformal to a pseudo-Euclidean space, invariant under the action of the pseudo-orthogonal group with zero scalar curvature is stady and consequently flat. How application of the results…

In this paper we show that an expanding or steady gradient Ricci soliton warped product $B^n\times_f F^m$, $m>1$, whose warping function $f$ reaches both maximum and minimum must be a Riemannian product. Moreover, we present a necessary and…

We prove structure results for homogeneous spaces that support a non-constant solution to two general classes of equations involving the Hessian of a function and an invariant 2-tensor. We also consider trace-free versions of these systems.…

We consider the class of compact n-dimensional Riemannian manifolds with cylindrical boundary, Ricci curvature bounded below by a given constant and injectivity radius bounded below by a positive constant, away from the boundary. For a…

In this paper we consider $M = B\times_{f}F$ warped product gradient Ricci solitons. We proved that the potential function depends only on the base and the fiber $F$ is necessarily Einstein manifold. We provide all such solutions in the…

We prove that, on any sub-Riemannian manifold endowed with a positive smooth measure, the Bakry-\'Emery inequality for the corresponding sub-Laplacian implies the existence of enough Killing vector fields on the tangent cone to force the…

In this paper, a systematic study of Kenmotsu pseudo-metric manifolds are introduced. After studying the properties of this manifolds, we provide necessary and sufficient condition for Kenmotsu pseudo-metric manifold to have constant…

We show that, up to biholomorphism, there is at most one complete $T^n$-invariant shrinking gradient K\"ahler-Ricci soliton on a non-compact toric manifold $M$. We also establish uniqueness without assuming $T^n$-invariance if the Ricci…

On an $n$-dimensional complete manifold $M$, consider an $h$-almost gradient Ricci soliton, which is a generalization of a gradient Ricci soliton. We prove that if the manifold is Bach-flat and $dh/du>0$, then the manifold $M$ is either…