Rectifiability
A Survey

A broad survey of the theory of rectifiability and its deep connections to numerous different areas of mathematics.

Pertti Mattila (Author)

9781009288088, Cambridge University Press

Paperback / softback, published 12 January 2023

182 pages
23 x 15.3 x 1 cm, 0.28 kg

Rectifiable sets, measures, currents and varifolds are foundational concepts in geometric measure theory. The last four decades have seen the emergence of a wealth of connections between rectifiability and other areas of analysis and geometry, including deep links with the calculus of variations and complex and harmonic analysis. This short book provides an easily digestible overview of this wide and active field, including discussions of historical background, the basic theory in Euclidean and non-Euclidean settings, and the appearance of rectifiability in analysis and geometry. The author avoids complicated technical arguments and long proofs, instead giving the reader a flavour of each of the topics in turn while providing full references to the wider literature in an extensive bibliography. It is a perfect introduction to the area for researchers and graduate students, who will find much inspiration for their own research inside.

Introduction
1. Preliminaries
2. Rectifiable curves
3. One-dimensional rectifiable sets
4. Higher dimensional rectifiable sets
5. Uniform rectifiability
6. Rectifiability of measures
7. Rectifiable sets in metric spaces
8. Heisenberg and Carnot groups
9. Bounded analytic functions and the Cauchy transform
10. Singular integrals
11. Harmonic measure and elliptic measures
12. Sets of finite perimeter and functions of bounded variation
13. Currents and varifolds
14. Minimizers and quasiminimizers
15. Rectifiability of singularities
16. Miscellaneous topics related to rectifiability
References
Index.

Subject Areas: Geometry [PBM], Complex analysis, complex variables [PBKD]