Séance Séminaire

Séminaire Gaston Darboux

Compactness of conformal metrics.

In this talk I will present two different results about compactness of conformal metrics. In the first part I will talk about relative isospectral surfaces and explain how to prove compactness in the smooth topology for sets of these surfaces (previous joint work with Pierre Albin and Frederic Rochon). In the second part, I will talk about recent joint work with Gilles Carron and Samuel Tapie. We consider metrics whose scalar curvatures are uniformly bounded in L^{n/2} and whose volumes are also uniformly bounded. The estimate in L^{n/2} is critical. However, this estimate implies that the volume density is a strong A infinity -weight. To be a strong A infinity -weight has important geometric consequences. In particular, for a sequence of conformal metrics, to have volume densities that are uniformly strong A infinity-weights, in addition to the volume bounds, gives compactness in the Gromov-Hausdorff topology. I will explain all the terminology and give all the necessary definitions.