Integral currents in metric manifolds, with applications to Poincaré inequalities

Contact: Stephan Stadler (stadler@mpim-bonn.mpg.de)

In the 1960s, Federer-Fleming developed their theory of normal and
integral currents in Euclidean space, providing a suitable setting to
study and solve Plateau's problem of finding area minimizing surfaces of
any dimension with prescribed boundary. Around 20 years ago,
Ambrosio-Kirchheim gave a vast generalization of Federer-Fleming's
theory to the setting of metric spaces. In particular, they introduced
integral currents in metric spaces, which can be thought of as measure
theoretic generalizations of oriented surfaces for which there are
natural notions of volume and boundary. In this talk we consider metric
spaces homeomorphic to a closed, orientable smooth manifold. We study
when such spaces (called metric manifolds) support a non-trivial
integral current without boundary. The existence of such an object
should be thought of as an analytic analog of the fundamental class of
the metric manifold. As an application, we obtain a conceptually simple
proof of a deep theorem of Semmes about the validity of a weak
1-Poincaré inequality in metric manifolds that are Ahlfors regular and
linearly locally contractible. In the smooth case the idea for this
simple proof goes back to Gromov. Poincaré inequalities are important in
the development of first order calculus in the setting of metric measure
spaces. Based on joint work with G. Basso and D. Marti.