On 4-dimensional gradient shrinking solitons

@article{Ni2007On4G,
  title={On 4-dimensional gradient shrinking solitons},
  author={Lei Ni and Nolan Wallach},
  journal={arXiv: Differential Geometry},
  year={2007}
}
  • Lei Ni, N. Wallach
  • Published 16 October 2007
  • Mathematics
  • arXiv: Differential Geometry
In this paper we classify the four dimensional gradient shrinking solitons under certain curvature conditions satisfied by all solitons arising from finite time singularities of Ricci flow on compact four manifolds with positive isotropic curvature. As a corollary we generalize a result of Perelman on three dimensional gradient shrinking solitons to dimension four. 

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References

SHOWING 1-10 OF 23 REFERENCES

Ricci solitons on compact three-manifolds

On a classification of the gradient shrinking solitons

TLDR
In high dimension this new method allows us to prove a classification result on gradient shrinking solitons with vanishing Weyl curvature tensor, which includes the rotationally symmetric ones.

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

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

Complete manifolds with nonnegative curvature operator

In this short note, as a simple application of the strong result proved recently by Bohm and Wilking, we give a classification on closed manifolds with 2-nonnegative curvature operator. Moreover, by

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

Manifolds with positive curvature operators are space forms

The Ricci flow was introduced by Hamilton in 1982 [H1] in order to prove that a compact three-manifold admitting a Riemannian metric of positive Ricci curvature is a spherical space form. In

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

Ricci Flow and the Poincare Conjecture

Background from Riemannian geometry and Ricci flow: Preliminaries from Riemannian geometry Manifolds of non-negative curvature Basics of Ricci flow The maximum principle Convergence results for Ricci