CCEM is a research project founded by Agence Nationale de la Recherche, 2018-2022.

Coordinator : Laurent Bessières

SYNOPSIS OF THE PROJECT
A fundamental problem in Riemannian geometry is to understand "spaces of metrics" satisfying variours curvature constraints. These spaces can be endowed with topologies, as the Gromov-Hausdorff one. When non compact it is natural to try to complete them by introducing singular metrics. This has led to the definition of several classes of singular metric spaces, studied for their links to Riemannian manifolds but also for themselves. Our project gather geometers specialists in topology, Ricci flow, analysis on manifolds and singular metrics spaces, with the aim to study these spaces of Riemannian or generalized metrics by combining our approaches and techniques. We envision questions of existence-uniqueness of "best metric" in a given class, of homotopy type of classes of metrics, generalisations of the theory of limits under Ricci bounds, as well as the study of some stratified spaces with conical iterated metrics.

Our team :
The project consortium is divided in four partners: Bordeaux (coord. Laurent Bessières), Grenoble (coord. Gérard Besson ), Montpellier (coord. Philippe Castillon) and Nantes (coord. Samuel Tapie ). The research institutions that will host our project are the Institut de Mathématiques de Bordeaux, Institut Fourier , Institut Montpelliérain Alexander Grothendieck and Laboratoire mathématique Jean Leray , respectively.
The other members of the team are Jérôme Bertrand (Toulouse), Gilles Carron (Nantes), Erwann Delay (Avignon), Alix Deruelle (UPMC), Gautier Dietrich (Montpellier), Marc Herzlich (Montpellier), Sylvain Maillot (Montpellier), Ilaria Mondello (UPEC), Thomas Richard (UPEC), Berardo Ruffini (Montpellier), Arnaud Stoker (Grenoble), Constantin Vernicos (Montpellier), Jian Wang (Grenoble).

Post-doctoral positions funded by the project will be available in Montpellier.

SYNOPSIS OF THE PROJECT
To begin with, let us explain some striking differences between closed and open manifolds. It is known that there are only countably many smooth closed n-manifolds, up to diffeomorphisms, and that there are uncountably many open n-dimensional manifolds (for n >1), even contractible ones (for n>2). Hence, almost all open manifolds do not cover closed ones. There are quite a few explicit examples of this situation among which the celebrated Whitehead 3-manifolds which are contractible open n-manifolds not homeomorphic to Rn. In dimension greater than 3, there are contractible open manifolds which are not homeomorphic to Rn but do cover compact manifolds. These remarks were chosen here only to stress the fact that open manifolds can appear in very different shapes. Hence the topology of open (i.e. noncompact) manifolds is much richer than that of compact manifolds. In addition to being interesting objects in their own right, open manifolds appear in the study of closed manifolds, for instance as limits of blow-ups of geometric flows or special sequences of Riemannian manifolds. The open manifolds appearing in these particular contexts are expected, and sometimes known, to have well behaved topology and geometry; therefore they deserve special attention.

In this project, we propose to study the topology of open 3-manifolds with the tools of Riemannian geometry. This idea has been highly successful in the compact case, one of the highlights being Perelman’s proof of the Poincaré and Geometrization Conjectures using Ricci flow, and we aim at broadening the scope of these techniques.

We will follow two approaches: in the first one, we intend to prove rigidity or classification results, where one obtains topological restrictions on a Riemannian manifold under certain geometric constraints. The second one consists in finding on a given open manifold a «nice» metric whose properties reflect the topology. Geometric flows, such as the well-known Ricci flow, or more recently introduced higher-order flows, are powerful analytic tools that can be used to tackle these questions. The study of these flows, in particular their singularities, relies in turn on a geometric and topological understanding of special sequences of Riemannian metrics.