Related papers: Vanishing bach-like tensors on complete gradient s…

We prove that there does not exist non-constant positive $f$-harmonic function on the complete gradient shrinking Ricci solitons. We also prove the $L^{p}(p\geq 1 \ {\rm or}\ 0<p\leq 1)$ Liouville theorems on the complete gradient shrinking…

We use analytical and numerical methods to investigate the equations for cohomogeneity one shrinking gradient Ricci solitons. We show the existence of a winding number for this system around the subvariety of phase space corresponding to…

A Ricci soliton $(M^n,g,v,\lambda)$ on a Riemannian manifold $(M^n,g)$ is said to have concurrent potential field if its potential field $v$ is a concurrent vector field. In the first part of this paper we completely classify Ricci solitons…

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…

We show that any tangent cone of a singular shrinking K\"ahler-Ricci soliton is a normal affine algebraic variety. Moreover, the regular set of such a tangent cone in the metric sense coincides with the regular set in the algebraic sense.…

We prove the vanishing of the first Betti number on compact manifolds admitting a Weyl structure whose Ricci tensor satisfies certain positivity conditions, thus obtaining a Bochner-type vanishing theorem in Weyl geometry. We also study…

Let us consider a generic n-dimensional subbundle V of the tangent bundle TM on some given manifold M. Given V one can define different degeneracy loci S_r(CV), r=(r_1<= r_2<= r_3<=...<=r_k) on M consisting of all points x in M for which…

In this paper we characterize the source-free Einstein-Maxwell spacetimes which have a trace-free Chevreton tensor. We show that this is equivalent to the Chevreton tensor being of pure-radiation type and that it restricts the spacetimes to…

We prove a vanishing and estimation theorem for the $p^{\text{th}}$-Betti number of closed $n$-dimensional Riemannian manifolds with a lower bound on the average of the lowest $n-p$ eigenvalues of the curvature operator. This generalizes…

In this paper, we give a description for steady Ricci solitons with a linear decay of sectional curvature. In particular, we classify all 3-dimensional steady Ricci solitons and 4-dimensional $\kappa$-noncollpased steady Ricci solitons with…

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…

In this paper we consider $4$-dimensional steady soliton singularity models, i.e., complete steady gradient Ricci solitons that arise as the rescaled limit of a finite time singular solution of the Ricci flow on a closed $4$-manifold. In…

In this article we classify expanding homogeneous Ricci solitons up to dimension 5, according to their presentation as homogeneous spaces. We obtain that they are all isometric to solvsolitons, and this in particular implies that the…

In this paper, we investigate left invariant Riemannian metrics on Lie groups with one and two-dimensional commutator subgroups. We explicitly provide the Levi-Civita connection, sectional curvature, and Ricci curvature, and we give…

Positive and negative flows of the Chen-Lee-Liu model and its various reductions, including Burgers hierarchy, are formulated within the framework of Riemann-Hilbert-Birkhoff decomposition with the constant grade two generator. Two classes…

We consider Ricci flow on two classes of nilpotent Lie groups that generalize the three-dimensional Heisenberg group: the higher-dimensional classical Heisenberg groups, and the groups of real unitriangular matrices. Each group is known to…

We show that a tensor field of any rank integrates to zero over all broken rays if and only if it is a symmetrized covariant derivative of a lower order tensor which satisfies a symmetry condition at the reflecting part of the boundary and…

This article explores Ricci-Yamabe solitons on a specific class of 4-dimensional Walker manifolds. Walker manifolds, characterized by the existence of a parallel null distribution, find applications in general relativity and are fundamental…

Almost Riemann solitons are introduced and studied on an almost contact complex Riemannian manifold, i.e. an almost contact B-metric manifold, obtained from a cosymplectic manifold of the considered type by a contact conformal…

In this article we have showed that a gradient $\rho$-Einstein soliton with a vector field of bounded norm and satisfying some other conditions is isometric to the Euclidean sphere. Later, we have proved that a non-trivial complete gradient…