In this paper, we consider the characterization of eigenfunctions for Laplacian operators on some Riemannian manifolds. Firstly we prove that for the space form (M K , gK) with the constant sectional curvature K, the first eigenvalue of Laplacian operator λ1 (M K) is greater than the limit of the first Dirichlet eigenvalue of Laplacian operator λ1 (BK (p, r)). Based on this, we then present a c...

In this paper we study *-Conformal {\eta}-Ricci soliton on Sasakian manifolds. Here, discuss some curvature properties manifold admitting soliton. We obtain significant results in manifolds satisfying R({\xi},X).S = 0, S({\xi},X).R {\overline}P({\xi},X).S where {\overline}P is Pseudo-projective tensor.The conditions for {\Phi}-conharmonically flat and {\Phi}-projectively have been obtained arti...

The subject of left-invariant Ricci soliton metrics on nilpotent Lie groups has enjoyed quite a bit of attention in the past several years. These metrics are intimately related to left-invariant Einstein metrics on non-unimodular solvable Lie groups. In fact, a classification of one is equivalent to a classification of the other. In this note, we focus our attention on nilpotent Lie groups and ...

In this paper, we first show that a complete shrinking breather with Ricci curvature bounded from below must be gradient soliton. This result has several applications. First, can classify all $3$-dimensional breathers. Second, every soliton -- generalization of Naber's result. Furthermore, develop general condition for the existence asymptotic soliton, which hopefully will contribute to study a...

We construct a 2-parameter family FZ of Riemannian metrics on the twistor space Z of a positive quaternion Kähler manifold M satisfying the following properties : (1) the family FZ contains an Einstein metric gZ and its scalings, (2) the family FZ is closed under the operation of making the convex sums, (3) the Ricci map g 7→ Ric(g) defines a dynamical system on the family FZ, (4) the Ricci flo...

In this paper we consider the dynamical system involved by the Ricci operator on the space of Kähler metrics. A. Nadel has defined an iteration scheme given by the Ricci operator for Fano manifold and asked whether it has some nontrivial periodic points. First, we prove that no such periodic points can exist. We define the inverse of the Ricci operator and consider the dynamical behaviour of it...

In this paper we consider the dynamical system involved by the Ricci operator on the space of Kähler metrics. A. Nadel has defined an iteration scheme given by the Ricci operator for Fano manifold and asked whether it has some nontrivial periodic points. First, we prove that no such periodic points can exist. We define the inverse of the Ricci operator and consider the dynamical behaviour of it...