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An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem. (English) Zbl 1138.53003
Progress in Mathematics 259. Basel: Birkhäuser (ISBN 978-3-7643-8132-5/hbk). xvi, 223 p. (2007).
The book is a very complete introduction to the topics cited in the title, beginning with the basis knowledge and motivations and reaching the most recent developments of the subject. It is clearly written and contains a lot of useful bibliographical references.
Chapter 1 contains a brief introduction to the planar isoperimetric problem, discussing all its different interpretations. Chapter 2 presents the basic properties of the Heisenberg group and of its invariant sub-Riemannian geometries. Chapter 3 deals with the links between the Heisenberg group and other mathematical structures. Chapter 4 studies the curvature-like properties of hypersurfaces in the Heisenberg group. Chapter 5 describes the properties of the Sobolev and BV spaces in Heisenberg geometry, focusing on embedding and Poincaré theorems. Chapter 6 is concerned with geometric measure theory in the Heisenberg groups. The chapter contains the proof of Pansu’s analogue of Rademacher’s theorem and, as a consequence, one of Mostow rigidity theorems for the complex hyperbolic plane. Chapter 7 presents the proof of Pansu’s isoperimetric inequality for the Heisenberg group. Chapter 8 explains Pansu’s conjecture on the isoperimetric subsets of the Heisenberg group and gives partial solutions. Chapter 9 deals with other geometric-analytic properties of the Heisenberg group.

53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
22E30 Analysis on real and complex Lie groups
53C17 Sub-Riemannian geometry