In this paper, we prove that Kähler-Ricci flow converges to a Kähler-Einstein metric (or a Kähler-Ricci soliton) in the sense of Cheeger-Gromov as long as an initial Kähler metric is very closed to gKE (or gKS) if a compact Kähler manifold with c1(M) > 0 admits a Kähler Einstein metric gKE (or a Kähler-Ricci soliton gKS). The result improves Main Theorem in [TZ3] in the sense of stability of Kä...

where k is the first Betti number b^M), T is a flat riemannian λ -torus, M~ is a compact connected Ricci-flat (n — λ;)-manifold, and Ψ is a finite group of fixed point free isometries of T x M' of a certain sort (Theorem 4.1). This extends Calabi's result on the structure of compact euclidean space forms ([7] see [20, p. 125]) from flat manifolds to Ricci-flat manifolds. We use it to essentiall...

The Ricci bracket flow is a geometric evolution on Lie algebras which is related to the Ricci flow on the corresponding Lie group. For nilpotent Lie groups, these two flows are equivalent. In the solvable case, it is not known whether they are equivalent. We examine a family of solvable Lie algebras and identify various elements of that family which are solitons under the Ricci bracket flow. We...

In this paper we prove a conjecture by Feldman–Ilmanen–Knopf (2003) that the gradient shrinking soliton metric they constructed on the tautological line bundle over CP is the uniform limit of blow-ups of a type I Ricci flow singularity on a closed manifold. We use this result to show that limits of blow-ups of Ricci flow singularities on closed four-dimensional manifolds do not necessarily have...

We describe the exponential map from an infinite-dimensional Lie algebra to an infinite-dimensional group of operators on a Hilbert space. Notions of differential geometry are introduced for these groups. In particular, the Ricci curvature, which is understood as the limit of the Ricci curvature of finite-dimensional groups, is calculated. We show that for some of these groups the Ricci curvatu...

Copyright q 2010 Zisheng Hu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We extend the classical Bishop-Gromov volume comparison from constant Ricci curvature lower bound to radially symmetric Ricci curvature lower bound...

We consider wormhole geometries subject to a gravitational action consisting of non-linear powers of the Ricci scalar. Specifically, wormhole throats are studied in the case where Einstein gravity is supplemented with a Ricci-squared and inverse Ricci term. In this modified theory it is found that static wormhole throats respecting the weak energy condition can exist. The analysis is done local...

Geometric monotone properties of the first nonzero eigenvalue of Laplacian form operator under the action of the Ricci flow in a compact nmanifold ( ) 2 ≥ n are studied. We introduce certain energy functional which proves to be monotonically non-decreasing, as an application, we show that all steady breathers are gradient steady solitons, which are Ricci flat metric. The results are also extend...

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...