Flots et Opérateurs Géométriques

ANR Programme blanc ANR-07-BLAN-0251-01

Résumé du projet (en anglais)

The technique of the Ricci flow was introduced in 1982 by R. Hamilton in a famous article. The aim of the method is to construct a deformation of a Riemannian metric on a compact manifold such that the evolution leads to a "beautiful" metric in the sense that it has nice properties such as being Einstein. In three dimensions an Einstein metric has constant curvature. R. Hamilton had in mind a proof of the Poincaré conjecture (and eventually the Geometrization conjecture) by constructing a constant curvature metric on a three-dimensional simply connected manifold. He could not achieve this goal but has nevertheless obtained deep results and has constructed a remarkable theory. Between december 2002 and july 2003, G. Perelman has posted three papers on the web in which he announced the proof of both conjectures. The texts are more a detailed sketch of proof rather than truly articles. Several groups started to work, everywhere in the world, in order to clarify the various details and write notes that could make the reading easier. In France it took some time for the geometer community to react and, to our knowledge, the only persons that went through the whole proof of the Poincaré conjecture are among the participants of this project. It is now commonly accepted that these two conjectures are proved. The following members of the project : L. Bessières, G. Besson, M. Boileau, S. Maillot and J. Porti (UAM, Bacelona, associated to the project), are writing a complete proof of the geometrization conjecture and are supported by the Clay Institute. It is worth mentioning that the main specialists of the technique are in the USA and to a lesser extent in China. The subject is less known in Europe except for some individuals. We aim at maintaining a state-of-the-art knowledge in France on this powerful and fruitful technique.

One of the goal of the project is to spread this revolutionary technique and its applications. In higher dimension, for exemple, the picture about the geometry and the topology is not clear, even in dimension 4. We aim to extend the techniques developped in dimension 3, associating other techniques introduced recently for the study of the topology and the geometry of four manifolds, namely higher order conformally covariant operators, Q-curvature, higher order flows associated to these invariants. Some of the participants of this project are specialists of this field, like P. Baird, Z. Djadli, A. Fardoun and R. Regbaoui. Among the activities which are already scheduled, we should mention the Borel trimester, co-organized by G. Besson, J. Lott and G. Tian, from april to june 2008, on the Ricci curvature with a strong component on the Ricci flow and its applications including, in particular, one workshop devoted to this theme. Let us recall that this trimester will be held exactly two years after the publication of the first set of notes on Perelman.s works and we hope that the decision made by the Clay Institute about the Clay prize (the million dollar) will be announced around this time. In july 2008, a workshop, organized in Brest by P. Baird will complete the activity. In 2009, there will be a three months program in Pisa on conformally covariant operators and higher order flows. These meetings will have a strong educational component. Indeed, various courses at different level will be organised so that graduate students could have enough background to attend the workshops. This is one of the reason why these workshops are late in the trimester (in june). This is one of the major preoccupation in our project: the project is meant to be the place for graduate students, postdoctoral fellows and young researchers (and also for people who want to become more familiar with the topics of the project) to participate actively to the activities we will organize, and to have fruitful interactions with senior researchers.