Ricci Flow with Surgery on Four-manifolds with Positive Isotropic Curvature

@article{Chen2005RicciFW,
  title={Ricci Flow with Surgery on Four-manifolds with Positive Isotropic Curvature},
  author={Binglong Chen and Xiping Zhu},
  journal={Journal of Differential Geometry},
  year={2005},
  volume={74},
  pages={177-264}
}
In this paper we study the Ricci flow on compact four-manifolds with positive isotropic curvature and with no essential incompressible space form. Our purpose is two-fold. One is to give a complete proof of Hamilton's classification theorem on four-manifolds with positive isotropic curvature and with no essential incompressible space form; the other is to extend some recent results of Perelman on the three-dimensional Ricci flow to four-manifolds. During the the proof we have actually provided… 
View PDF on arXiv
75 Citations
Highly Influential Citations
10
Background Citations
25
Methods Citations
18
Results Citations
4

75 Citations

Citation Type

Citation Type

Isotropic Curvature and the Ricci Flow
In this paper, we study the Ricci flow on higher dimensional compact manifolds. We prove that nonnegative isotropic curvature is preserved by the Ricci flow in dimensions greater than or equal to…
Four-manifolds with positive isotropic curvature
We give a survey on 4-dimensional manifolds with positive isotropic curvature. We will introduce the work of B. L. Chen, S. H. Tang and X. P. Zhu on a complete classification theorem on compact…
Ricci flow with surgery on manifolds with positive isotropic curvature
We study the Ricci flow for initial metrics with positive isotropic curvature (strictly PIC for short). In the first part of this paper, we prove new curvature pinching estimates which ensure that…
Manifolds with 1/4-pinched curvature are space forms
Let (M,g_0) be a compact Riemannian manifold with pointwise 1/4-pinched sectional curvatures. We show that the Ricci flow deforms g_0 to a constant curvature metric. The proof uses the fact, also…
Noncompact 4-manifolds with uniformly positive isotropic curvature
In this paper, we study Ricci flow on noncompact 4-manifolds with uniformly positive isotropic curvature and with no essential imcompressible space form. That means there is positive lower bound of…
Positive isotropic curvature and self-duality in dimension 4
We study a positivity condition for the curvature of oriented Riemannian 4-manifolds: the half-PIC condition. It is a slight weakening of the positive isotropic curvature (PIC) condition introduced…
A sphere theorem for three dimensional manifolds with integral pinched curvature
In a previous paper, we proved a number of optimal rigidity results for Riemannian manifolds of dimension greater than four whose curvature satisfy an integral pinching. In this article, we use the…
Uniqueness of the Ricci flow on complete noncompact manifolds
The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton \cite{Ha1}. Later on,…
Ricci flow with surgery in higher dimensions
We present a new curvature condition which is preserved by the Ricci flow in higher dimensions. For initial metrics satisfying this condition, we establish a higher dimensional version of Hamilton's…
Exotic R4's and positive isotropic curvature
...
...

References

SHOWING 1-10 OF 50 REFERENCES
Ricci flow with surgery on three-manifolds
This is a technical paper, which is a continuation of math.DG/0211159. Here we construct Ricci flow with surgeries and verify most of the assertions, made in section 13 of that e-print: the…
The structure of complete manifolds of nonnegative curvature
0. In this paper we describe some results on the structure of complete manifolds of nonnegative sectional curvature. (We will denote such manifolds by M.) Details and related results will appear…
Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes
There has been much interest among differential geometers in finding relationships between curvature and topology of Riemannian manifolds. For the most part, efforts have been directed towards…
The Ricci flow on the 2-sphere
The classical uniformization theorem, interpreted differential geomet-rically, states that any Riemannian metric on a 2-dimensional surface ispointwise conformal to a constant curvature metric. Thus…
Uniqueness of the Ricci flow on complete noncompact manifolds
The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton \cite{Ha1}. Later on,…
Deforming metrics in the direction of their Ricci tensors
In [4], R. Hamilton has proved that if a compact manifold M of dimension three admits a C Riemannian metric g0 with positive Ricci curvature, then it also admits a metric g with constant positive…
The entropy formula for the Ricci flow and its geometric applications
We present a monotonic expression for the Ricci flow, valid in all dimensions and without curvature assumptions. It is interpreted as an entropy for a certain canonical ensemble. Several geometric…
A Uniformization Theorem For Complete Non-compact Kähler Surfaces With Positive Bisectional Curvature
In this paper, by combining techniques from Ricci flow and algebraic geometry, we prove the following generalization of the classical uniformization theorem of Riemann surfaces. Given a complete…
Four-manifolds with positive isotropic curvature
1. Positive Isotropic Curvature 2 (1) The Result 2 (2) The Algebra of Isotropic Curvature 4 2. Curvature Pinching 6 (1) Pinching Estimates which are Preserved 6 (2) Pinching Estimates which Improve…
Ricci deformation of the metric on a Riemannian manifold
The interaction between algebraic properties of the curvature tensor and the global topology and geometry of a Riemannian manifold has been studied extensively. Of particular interest is the question…
...
...