Organisation: Prof. Werner Ballmann, Stéphane Félix

Ort und Zeit: Freitag, um 14:15, Seminarraum D, ab 17. Oktober 2008.
Nächsten Vortrag: 23.1.09, 14:15

Inhalt: Im Fall von Mannigfaltigkeiten ist die Theorie der Ströme (vorgestellt in [2]) eine Verallgemeinerung der Distributionentheorie. Ambrosio und Kirchheim [1] haben eine Theorie der Ströme auf metrische Räumen entwickelt. Die Grundidee hiervon ist, den im klassischen Fall benutzten Raum von Testfunktionen (i.e. Differentialformen mit kompaktem Träger) durch einen Raum von Lipschitzfunktionen zu ersetzen. Lang [3] greift diese Theorie mit reduzierten Voraussetzungen auf. Insbesondere benötigt man anstelle endlicher Maße für Ströme nur noch lokal endliche Maße. Ziel des Seminars ist, die Arbeit von Lang durchzuarbeiten. Zentrale Begriffe sind: Rand, Push-forward, Maße von Strömen, normale und integrale Ströme, slicing.

Content: In the case of manifolds, the theory of currents (as presented in [2] for example) is a generalised theory of distributions. Ambrosio and Kirchheim [1] have developed a theory of currents on metric spaces. Roughly said, the start idea of this theory is to replace the space of test functions used in the classical theory (that is, differential forms with compact support) by a well-chosen space of lipschitz functions. Lang [3] writes down this theory with reduced assumptions. In particular, it is not necessary to suppose from the beginning that the currents have finite mass, but only locally finite mass. The goal of the seminar is to study the article of Lang. The central notions are: boundary, push-forward, mass of a current, normal and integral currents, and slicing.

Program:

1. (Stéphane Félix, 17.10.08) Classical theory of currents, for ex. in [5]: differential forms, currents, boundary, mass, push-forward, restriction, integral representation, compactness theorems, examples. From section 1 of article [3]: Lebesgue points, smoothing.
2. (Robert Nabiullin, 14.11.08) Section 2 of article [3]: Definition of metric currents, topology, restriction, product and chain rules, example of a current induced by a function.
3. (Steffen Weil, 21.11.08) Section 3: support, boundary, push-forward, formula of Lemma 3.5
4. (Lara Skuppin, 28.11.08) Section 4: total mass and mass of a current, completeness of the space of finite mass, properties of mass, restriction of a current on a Borel set.
5. (Shehryar Sikander, 12.12.08) Section 5: normal currents, uniform continuity and compactness theorem for normal currents, flat norm and comparison map to euclidean spaces.
6. (Blanka Horvath, 19.12.08) Section 6: Definition of the slicing of a normal current by one or more lipschitz maps, properties, continuity questions.
7. (Oeznur Albayrak, 16.1.09) Section 7: functions with bounded variation, Theorem 7.6.
8. (Johannes Klimpt, 23.1.09) Section 8: rectifiability of a current, integral currents, compactness theorem.

References:

• [1] Luigi Ambrosio, Bernd Kirchheim, Currents in metric spaces, Acta Math., vol. 185, 2000
• [2] Herbert Federer, Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969
• [3] Urs Lang, Local currents in metric spaces, Preprint, 2008, available here.
• [4] Frank Morgan, Geometric measure theory. A beginner's guide. Third edition. Academic Press, Inc., San Diego, CA, 2000.
• [5] Urs Lang, Introduction to Geometric Measure Theory (lecture notes), available here.