Université Paris-Est Université Paris-Est - Marne-la-Vallée Université Paris-Est - Créteil Val-de-Marne Centre National de la Recherche Scientifique

The Plateau problem in metric spaces

21/05/2012 - 14:00 - 15:00
HARDT Robert
Rice University

Around 1960 (in work of DeGiorgi, Federer-Fleming, Reifenberg) questions concerning the existence, in $\mathbb{R}^n$ (or in a smooth Riemannian manifold) of mass minimizing geometric objects of dimension greater than $2$ led to consideration of various weak limits of submanifolds and stimulated the growth of geometric measure theory. Such minimizers, either having a given boundary or a given homology class, turned out not necessarily to be entirely smooth manifolds. But they only missed by some lower dimensional closed sets (whose structuree is still actively researched).

Suppose the ambient space itself is not a smooth Riemannian manifold. In 2000 Ambrosio and Kirchheim described "Currents in a metric space" to address such Plateau-type problems in a general metric space. A joint work with De Pauw generalizes this and provides chains producing many homology groups involving the metric. Here we also discuss another current work with De Pauw and Pfeffer on real cochains called "charges" that are topologized variationally and that give a geometric cohomology. Duality problems lead to looking for spaces that enjoyi a the linear isoperimetric inequality. There are potentially interesting applications with the ambient metric space being a singular variety or a fractal or a metric limit of Riemannian manifolds.