Ricci flow and geometrization

Lecturer

Christoph Boehm (Munster)

Contents

Let (M,g) be a smooth Riemannian manifold. The Ricci flow is the geometric evolution equation

$\frac{\partial}{\partial t}\,g(t)=-2\textrm{Ric}(g(t))\,,\quad g(0)=g$

which was introduced by Richard Hamilton in 1982.

One hopes that the geometry of a solution g(t) of the Ricci flow improves so that one can see the topology of the underlying manifold. For instance, in 1986 Hamilton proved that on a closed surface, the normalized Ricci flow converges to a metric of constant curvature.

In this course, we will present some of Hamilton's fundamental work on Ricci flow, such as short time existence, Hamilton's maximum principle for tensors, higher derivative estimates and long time existence. As an application one obtains the classification of closed 3-manifolds of positive Ricci curvature given by Hamilton in 1982.

Furthermore, we will describe some results of Perelman's spectacular work on Ricci flow, such as Perelman's entropy functional and the Noncollapsing Theorem. Then we will indicate how these methods can be used to approach Thurston's Geometrization Conjecture for 3-manifolds.

Literature

• R. Hamilton, "Three-manifolds with positive Ricci curvature", J. Diff. Geometry (17), 255-306 (1982).
• R. Hamilton, "The formation of singularities in the Ricci flow", Surveys in Differential Geometry, Internat. Press, Cambridge, 1995.
• G. Perelman, "The Entropy Formula for the Ricci Flow and its Geometric Applications", arXiv:math.DG/0211159.
• B. Chow, D. Knopf, "The Ricci flow: An introduction", Math. Surveys and Monographs, Volume 110, AMS.

Schedule

The course will run during the first spring quater (30/1 - 17/3) of 2006 with 2 hours of lectures Tuesdays and Thursdays.

Financial support

Some amount of funding is avaible as support for travel expenses to Aarhus in connection with participation in (parts of) this course. Please contact the director of the center, Jørgen Ellegaard Andersen at for more information.

The course is partially funded by "Forskerskole i Matematik og anvendelser" based at the Department of Mathematics, University of Copenhagen.