This will take place on Thursday February 18th starting at 9am in CYCL07 (Marc de Hemptinne). The afternoon talks will be delivered in CYCL04.
ABSTRACT:
The aim of these lectures is to study a new distance between Riemannian manifolds and, more generally, countably m-rectifiable metric spaces. The definition of this "intrinsic flat" distance is inspired by that of the Gromov-Hausdorff distance but uses the flat distance from geometric measure theory rather than the Hausdorff distance. One advantage of working with this rather than the Gromov-Hausdorff distance is that intrinsic flat limits are "nicer" than Gromov-Hausdorff limits and sequences converge under weaker assumptions. On the other hand, due to a cancellation phenomenon, more "information" can be lost in the intrinsic flat limit than in the Gromov-Hausdorff limit. We will exhibit sufficient conditions on the topology of the spaces in a sequence which ensure that cancellation does not occur and that the intrinsic flat limit agrees with the Gromov-Hausdorff limit. This allows us for example to prove countable m-rectifiability of the Gromov-Hausdorff limit of certain sequences of Riemannian manifolds. We finally describe a new compactness theorem for intrinsic flat convergence, akin to Gromov's famous compactness theorem for sequences of metric spaces. In contrast to Gromov's theorem, which assumes uniform compactness of the sequence, our theorem only assumes uniform bounds on diameter and volume. Applications of this compactness theorem to isoperimetric inequalities for higher-dimensional cycles are discussed at the end. The lectures, which will be self-contained, are partly based on joint work with C. Sormani and R. Schul.
ROUGH OUTLINE OF THE LECTURES:
- Overview and rough description of results, for simplicity in the setting of Riemannian manifolds
- Ambrosio-Kirchheim's theory of integral currents in metric spaces
- Weak convergence of integral currents versus Hausdorff convergence of their supports
- Definition of the intrinsic flat distance, examples, relationship between Gromov-Hausdorff and intrinsic flat convergence; Gromov-Hausdorff limits of manifolds with Ricci curvature bounded from below
- Compactness theorem for intrinsic flat convergence and applications to isoperimetric problems
