Introduction to Geometric Measure Theory - Spring 2019
Lecturer Prof. S. Wenger

Start date
Mondays 13:15 - 15:00 in Science de la terre 1.309 and Fridays 10:15-12:00 in Physics 2.52

18 February 2019
The following problem, called Plateau's problem, lies at the origins of Geometric Measure Theory: Does every Jordan curve bound a surface of minimal area? The name goes back to the Belgian physicist Joseph Plateau who made extensive experiments with soap films in order to find an answer to this problem.

One of the principal achievements of Geometric Measure Theory has been to develop a sufficiently rich and powerful theory of surfaces which can be used to solve this problem and many related geometric variational problems.

This course provides an introduction to Geometric Measure Theory. Prerequisites are the course Mesure et Intégration (MA.3400/4400) and a good understanding of submanifolds and differential forms. A course on Functional Analysis is recommended but not strictly required.

Outline of course (might change slightly as we go along):

A) Review of measure theory:
  1. Outer measures
  2. Hausdorff measure and dimension
  3. Riesz representation theorem
  4. Weak compactness of Radon measures and Banach-Alaoglu theorem
B) Lipschitz maps:
  1. Lipschitz extensions
  2. Rademacher's theorem
  3. Area and co-area formulas
  4. Rectifiable sets
C) Review of differential forms
  1. m-vectors and m-covectors
  2. Differential forms and Stokes' theorem
D) Theory of currents:
  1. Definition and basic examples
  2. Homotopy formula and push-forward
  3. Integer rectifiable and integral currents
  4. Slicing
  5. Proofs of Closure and Boundary Rectifiability Theorems
  6. MBV functions and proof of the slice-rectifiability theorem

Preliminary notes for the course.

  • L. Simon: Lectures on Geometric Measure Theory. Australian National University.
  • S. Krantz, H. Parks: Geometric Integration Theory, Birkhaeuser.
  • F. Morgan: Geometric Measure Theory: A beginner's guide, Elsevier/Academic Press.
  • L. Evans, R. Gariepy: Measure theory and fine properties of functions, CRC Press.
  • P. Mattila: Geometry of Sets and Measures in Euclidean Spaces, Cambridge University Press.
  • L. Ambrosio, P. Tilli: Topics on Analysis in Metric Spaces, Oxford Lecture Series in Mathematics and its Applications.
  • H. Federer: Geometric Measure Theory, Springer Verlag.
  • L. Ambrosio, B. Kirchheim: Currents in metric spaces, Acta Math. 185 (2000), 1 - 80.