We classify the Kähler metrics on compact manifolds of complex dimension two that are solitons for the constant-volume Ricci flow, assuming that the curvature is slightly more positive than that of the single known example of a soliton in this dimension.

We study complete noncompact long time solutions (M, g(t)) to the Kähler-Ricci flow with uniformly bounded nonnegative holomorphic bisectional curvature. We will show that when the Ricci curvature is positive and uniformly pinched, i.e. Ri̄ ≥ cRgi̄ at (p, t) for all t for some c > 0, then there always exists a local gradient Kähler-Ricci soliton limit around p after possibly rescaling g(t) alon...

In this note we provide a proof of the following: Any compact KRS with positive bisectional curvature is biholomorphic to the complex projective space. As a corollary, we obtain an alternative proof of the Frankel conjecture by using the Kähler-Ricci flow. The purpose of this note is to give a proof of the following theorem, which does not rely on the previous solutions of Frankel conjecture: T...

The goal of the paper is to deliberate conformal Ricci soliton and *-conformal within framework paracontact geometry. Here we prove that if an ?-Einstein para-Kenmotsu manifold admits soliton, then it Einstein. Further have shown 3-dimensional para-cosymplectic flat satisfies where vector field conformal. We also constructed some examples verify our results.

The purpose of the present expository paper is to give an account of the recent progress and present status of the classification of solvable Lie groups admitting an Einstein left invariant Riemannian metric, the only known examples so far of noncompact Einstein homogeneous manifolds. The problem turns to be equivalent to the classification of Ricci soliton left invariant metrics on nilpotent L...

This note surveys and compares results in [12] and [21, 22] on the separation of variables construction for soliton solutions of curvature equations including the Kähler-Ricci flow and the Lagrangian mean curvature flow. In the last section, we propose some new generalizations in the Lagrangian mean curvature flow case.

In this article we have proved that a gradient Yamabe soliton satisfying some additional conditions must be of constant scalar curvature. Later, showed in expanding or steady with non-negative Ricci curvature if the potential function satisfies integral condition then it is subharmonic, particular, for case becomes harmonic. Also that, compact soliton, agrees Hodge-de Rham upto constant.

In this article, we have proved some results in connection with the potential vector field having finite global norm quasi Yamabe soliton. We derived criteria for on non-positive Ricci curvature of Also, a necessary condition compact soliton has been formulated. further showed that if complete non-trivial, non-compact volume, then scalar becomes constant and reduces to

The author proves that the isoperimetric inequality on graphic curves over circle or hyperplanes \({\mathbb{S}^{n - 1}}\) is satisfied in cigar steady soliton and Bryant soliton. Since both of them are Riemannian manifolds with warped product metric, utilize result Guan-Li-Wang to get his conclusion. For sake structure, believes geometric restrictions for which holds naturally Ricci solitons.

In this paper, we consider $\left(LCS\right)_{n}$ manifold admitting almost $\eta-$Ricci solitons by means of curvature tensors. Ricci pseudosymmetry concepts soliton have introduced according to the choice some special tensors such as pseudo-projective, $W_{1}$, $W_{1}^{\ast}$ and $W_{2}.$ Then, again tensor, necessary conditions are searched for be semisymmetric. Then characterizations obtain...