The formation of singularities in the Ricci flow.

*(English)*Zbl 0867.53030
Hsiung, C. C. (ed.) et al., Proceedings of the conference on geometry and topology held at Harvard University, Cambridge, MA, USA, April 23-25, 1993. Cambridge, MA: International Press. Surv. Differ. Geom., Suppl. J. Differ. Geom. 2, 7-136 (1995).

The article provides a rather complete and thorough survey of geometrical properties of one-parameter families \((M(t),g(t))\) of Riemannian manifolds if the evolution of the metric is governed by the equation \(\partial g/\partial t=-2\text{ Ric }g\) (which can be identified with the heat-type equation \(\partial g/\partial t=\Delta g\) in appropriate local coordinates), especially to the subtle analysis of existence theorems, including numerous kinds of estimates and the mechanism of formation of various singularities. The singularities are special and therefore cannot occur in certain cases which may give the existence of the flow. On the other hand, continuing the flow through the singularity may change the topology of the manifold \(M(t)\) in a useful manner.

The paper can be read with minimal prerequisities and the topics are as follows. Examples of exact solutions, intuitive aspects, evolution of curvature with derivatives estimates, existence and uniqueness theorems, the convergence to an Einstein metric, the case of Kähler manifolds, evolution of geodesics and of minimal surfaces, the injectivity radius of a metric at a point, the change of the distance, the geometry at infinity (the aperture of the manifold), the behaviour of the ancient solutions (where \(-\infty<t<\text{const}\).), the structure of solitons (solutions stationary modulo diffeomorphisms), the influence of a bump for strictly positive curvature (with a review of the Topogonov theorem), dimension reduction, isoperimetric bounds and curvature pinching in three dimensions.

For the entire collection see [Zbl 0835.00021].

The paper can be read with minimal prerequisities and the topics are as follows. Examples of exact solutions, intuitive aspects, evolution of curvature with derivatives estimates, existence and uniqueness theorems, the convergence to an Einstein metric, the case of Kähler manifolds, evolution of geodesics and of minimal surfaces, the injectivity radius of a metric at a point, the change of the distance, the geometry at infinity (the aperture of the manifold), the behaviour of the ancient solutions (where \(-\infty<t<\text{const}\).), the structure of solitons (solutions stationary modulo diffeomorphisms), the influence of a bump for strictly positive curvature (with a review of the Topogonov theorem), dimension reduction, isoperimetric bounds and curvature pinching in three dimensions.

For the entire collection see [Zbl 0835.00021].

Reviewer: J.Chrastina (Brno)

##### MSC:

53C20 | Global Riemannian geometry, including pinching |

58J60 | Relations of PDEs with special manifold structures (Riemannian, Finsler, etc.) |

58J35 | Heat and other parabolic equation methods for PDEs on manifolds |

58D25 | Equations in function spaces; evolution equations |

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |