Topics in Geometric Measure Theory: Lipschitz Functions and Rectifiable Sets

DATES: April 6th - 8th and 11th - 13th, 2022 (6 sessions)
TIME: 9:30-11:30 (a total of 12 hours)
LOCATION: BCAM and Online

ABSTRACT:
This course aims at providing an introduction to the theory of rectifiable sets/measures, which is one of the central concepts in geometric measure theory (GMT). Loosely speaking, rectifiable sets are the sets with the least possible regularity required to admit approximate tangent planes, and they can be considered as the measure theoretical generalization of smooth manifolds. It is important because in many geometric/analytical problems, the object of study (surfaces, maps, support of measures, etc.) may not be smooth and rectifiability just gives the right amount of differentiable structure for the result to hold. Consequently, it has numerous applications in problems arising in the calculus of variations, geometric analysis, nonlinear partial differential equations, harmonic analysis, etc.

In this course, we first give an overview of basic notions and facts in measure theory and classical theorems for Lipschitz functions. After that we define the rectifiable set/measure and provide some basic properties. Then we will focus on various criteria of rectifiability through tangent measures, densities and projection properties, which will take the majority of the time of this course. The deep theorems, such as Marstrand-Mattila rectifiable theorem, Besicovitch-Federer projection theorem, Preiss's theorem, etc. will be covered. Besides the theory, some examples and applications will also be mentioned in the lectures. To finish, we will briefly introduce the rectifiable Reifenberg theorem by Naber & Valtorta, which recently has a wide application in various geometric variational problems.

PROGRAMME:
1. Preliminaries.
a. Basic notions of Hausdorff measures, covering theorems, density properties of sets, tangent measures, Marstrand's theorem.
b. Classical theorems for Lipschitz functions, including the extension theorem, Rademacher's differentiability theorem and Whitney's extension theorem.
c. Definition of rectifiable sets/measures. Area formula.
2. Rectifiability Criteria
a. The classical characterization of rectifiable sets in terms of the existence of an approximate tangent plane almost everywhere.
b. The Marstrand-Mattila rectifiability theorem under the weak tangent plane properties.
c. The Besicovitch-Federer projection theorem which characterizes rectifiable sets by their projection properties.
d. Discussion of the Preiss's deep theorem on density and rectifiability.
e. The rectifiable Reifenberg theorem and its applications to various geometric variational problems (harmonic maps, liquid crystal, minimal surface)

REFERENCES:
[1] C. De Lellis. Rectifiable sets, densities and tangent measures. Zurich Lectures in Advanced Mathematics. European Mathematical Society (EMS), Zúrich, 2008.
[2] H. Federer. Geometric measure theory. series Die Grundlehren der mathematischen Wissenschaften, Band 153, Springer-Verlag New York Inc., New York, 1969.
[3] P. Mattila. Geometry of sets and measures in Euclidean spaces. volume 44 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1995.
[4] F. Lin, X. Yang. Geometric measure theory: An introduction. International Press; 2003.
[5] L. Ambrosio, N. Fusco, D. Pallara. Functions of bounded variations and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000.
[6] D. Preiss. Geometry of measures in R^n: distribution, rectifiability, and densities. Ann. of Math. 125 (1987), 537-643.
[7] A. Naber, D. Valtorta. Rectifiable-Reifenberg and the Regularity of Stationary and Minimizing Harmonic Maps. Ann. of Math. 185 no.1 (2017), 131-227.


*Registration is free, but mandatory before March 30th. To sign-up go to https://forms.gle/gY3D43DG1odeNQAD9 and fill the registration form.

• BCAM-Course_20220406