Introduction to rectifiable currents

In this class I will present the theory of rectifiable currents. These are a measure theoretic analog of the notion of smooth submanifolds of Riemannian manifolds, and emerge in many applications in geometric analysis and mathematical physics. After an introduction concerning the general aspects of the topic and the basic definitions, the course will cover:

1.  the Federer and Fleming's theory, reviewed in the "metric space'' approach by Ambrosio and Kirchheim;
2.   the regularity of codimension 1 mass minimizing currents;
3.   the analysis of singular cones, and consequent dimension reduction arguments for the singular set;
4.   (time permitting) the boundary regularity of codimension 1 mass minimizing currents, after Hardt and Simon.

The class will be complemented by an exercise section and a reading seminar. The participants will be asked to actively take part to the course by solving some exercise sheets (which will integrate the lectures) and giving a seminar on a previously assigned paper on related topics.

References

•   L. Simon, Lecture notes on Geometric Measure Theory.
•   F. Lin and X. Yang, Geometric Measure Theory. An introduction.
•   S. Krantz and H. Parks, Geometric Integration Theory.

Date and time info
Tuesday 10.00 - 12.00

Keywords
Minimal Surfaces, Geometric Measure Theory, Regularity Theory of nonlinear PDEs

Prerequisites
This is an advanced class in Geometric Measure Theory. It is assumed a good knowledge about differential geometry, measure theory, functional analysis and partial differential equations. The basic results and the techniques borrowed by these fields will be every time suitably introduced and discussed, but no proof will be provided.

Language
English