Rectifiability: A Survey

Pertti Mattila

doi:10.1017/9781009288057

Rectifiability

A Survey

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Bortz, S. Poggi, B. Tapiola, O. and Tolsa, X. 2024. The $$A_\infty $$ Condition, $$\varepsilon $$-Approximators, and Varopoulos Extensions in Uniform Domains. The Journal of Geometric Analysis, Vol. 34, Issue. 7,

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Tapiola, Olli and Tolsa, Xavier 2024. Connectivity conditions and boundary Poincaré inequalities. Analysis & PDE, Vol. 17, Issue. 5, p. 1831.

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Antonelli, Gioacchino Brena, Camillo and Pasqualetto, Enrico 2024. The rank-one theorem on RCD spaces. Analysis & PDE, Vol. 17, Issue. 8, p. 2797.

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Pertti Mattila, University of Helsinki

Publisher:: Cambridge University Press
Online publication date:: December 2022
Print publication year:: 2023
Online ISBN:: 9781009288057
DOI:: https://doi.org/10.1017/9781009288057
Creative Commons:: This content is Open Access and distributed under the terms of the Creative Commons Attribution licence CC-BY-NC 4.0 https://creativecommons.org/creativelicenses

Subjects:: Mathematics (general), Mathematics, Real and Complex Analysis, Geometry and Topology
Series:: London Mathematical Society Lecture Note Series (483)

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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.

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Rectifiability

A Survey

This Book has been cited by the following publications. This list is generated based on data provided by Crossref.

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Contents

Frontmatter
pp i-iv

Contents
pp v-viii

Introduction
pp 1-2

1 - Preliminaries
pp 3-5

2 - Rectifiable Curves
pp 6-8

3 - One-Dimensional Rectifiable Sets
pp 9-17

4 - Higher-Dimensional Rectifiable Sets
pp 18-36

5 - Uniform Rectifiability
pp 37-46

6 - Rectifiability of Measures
pp 47-51

7 - Rectifiable Sets in Metric Spaces
pp 52-62

8 - Heisenberg and Carnot Groups
pp 63-77

9 - Bounded Analytic Functions and the Cauchy Transform
pp 78-88

10 - Singular Integrals
pp 89-98

11 - Harmonic Measure and Elliptic Measures
pp 99-106

12 - Sets of Finite Perimeter and Functions of Bounded Variation
pp 107-114

13 - Currents and Varifolds
pp 115-124

14 - Minimizers and Quasiminimizers
pp 125-128

15 - Rectifiability of Singularities
pp 129-142

16 - Miscellaneous Topics Related to Rectifiability
pp 143-148

References
pp 149-170

Index
pp 171-172

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Rectifiability

A Survey

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