In this work we construct and analyze exact solutions describing Ricci flows and nonholonomic deformations of four dimensional (4D) Taub-NUT spacetimes. It is outlined a new geometric techniques of constructing Ricci flow solutions. Some conceptual issues on spacetimes provided with generic off–diagonal metrics and associated nonlinear connection structures are analyzed. The limit from gravity/...

In this work we construct and analyze exact solutions describing Ricci flows and nonholonomic deformations of four dimensional (4D) Taub-NUT spacetimes. It is outlined a new geometric techniques of constructing Ricci flow solutions. Some conceptual issues on spacetimes provided with generic off–diagonal metrics and associated nonlinear connection structures are analyzed. The limit from gravity/...

In this note, we study a Kähler-Ricci flow modified from the classic version. In the non-degenerate case, strong convergence at infinite time is achieved. The main focus should be on degenerate case, where some partial results are presented. 1 Set-up and Motivation Kähler-Ricci flow, which is nothing but Ricci flow with initial metric being Kähler, enjoys the same debut as Ricci flow in R. Hami...

The notion of Ricci solitons was introduced by Hamilton [24] in mid 1980s. They are natural generalizations of Einstein metrics. Ricci solitons also correspond to self-similar solutions of Hamilton’s Ricci flow [22], and often arise as limits of dilations of singularities in the Ricci flow. In this paper, we will focus our attention on complete gradient shrinking Ricci solitons and survey some ...

We investigate the limiting behavior of the unnormalized Kähler-Ricci flow on a Kähler manifold with a polarized initial Kähler metric. We prove that the Kähler-Ricci flow becomes extinct in finite time if and only if the manifold has positive first Chern class and the initial Kähler class is proportional to the first Chern class of the manifold. This proves a conjecture of Tian for the smooth ...

In this paper we will give a rigorous proof of the lower bound for the scalar curvature of the standard solution of the Ricci flow conjectured by G. Perelman. We will prove that the scalar curvature R of the standard solution satisfies R(x, t) ≥ C0/(1−t) ∀x ∈ R , 0 ≤ t < 1, for some constant C0 > 0. Recently there is a lot of study of Ricci flow on manifolds by R. Hamilton [H1-6], S.Y. Hsu [Hs1...

The aim of this project is to introduce the basics of Hamilton’s Ricci Flow. The Ricci flow is a pde for evolving the metric tensor in a Riemannian manifold to make it “rounder”, in the hope that one may draw topological conclusions from the existence of such “round” metrics. Indeed, the Ricci flow has recently been used to prove two very deep theorems in topology, namely the Geometrization and...

d dt g = −2Rc g(0) = g0 on a closed manifold M. An important ingredient of Perelman’s proof of geometrization conjecture is the non-collapsing theorem of Ricci flow which makes sure that we can get a singularity model of the flow when a singularity exists. In [7], Zhang gave an easier way to prove the noncollapsing theorem of Ricci flow via a uniform Sobolev inequality along the flow. Unfortuna...