Project A5 of SFB/Transregio 71: Minimizing Normal Currents and the Stable Norm.

What is the minimal volume (of representatives) of a homology class in a compact Riemannian manifold and what are the properties of minimizing representatives? These are natural questions that can be asked also for more general parametric variational integrands instead of Riemannian volume. The answers depend on the choice of coefficients for the homology. In this project, we are interested in the case of real coefficients. The minimal volume of real homology classes defines a norm on the homology vector spaces, called stable norm. Geometric measure theory provides a natural notion of weak solution to the problem of finding minimal representatives - namely closed normal currents. Our general aim is to study such minimizing closed normal currents. In turn, their properties reveal features of the stable norm, such as facial structure and differtiability properties of the stable norm ball. The results will depend strongly on the topology and geometry of the manifold, but also fundamentally on the dimension q of the homology involved. While the case q=1 can be seen as a geometric case of Aubry-Mather's theory of action-minimizing orbits, we are presently working on the codimension one case, q=dim M -1. In this case, minimizing closed normal currents are measured laminations by minimizing hypersurfaces (possibly with a small singular set). A longterm project is to study minimal closed normal currents and the stable norm also in the intermediate dimensions 1 < q < dim M -1.

Additional information: http://sfbtr71.de/
contact person: Prof. Dr. Victor Bangert
Phone: 0761/203-5562
Email: bangert@email.mathematik.uni-freiburg.de

Start of project: 2009
End of project: 2013

Albert-Ludwigs-University Freiburg
Prof. Dr. Victor Bangert
Abteilung für Reine Mathematik
Prof. Dr. Victor Bangert
Eckerstraße 1
79104 Freiburg
Germany

Phone: +49 761 203-5562
Fax: +49 761 203-5541
Email: bangert@email.mathematik.uni-freiburg.de
http://home.mathematik.uni-freiburg.de/geometrie/bangert/
Actual Research Report

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