An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem (Progress in Mathematics, 259) - Hardcover

Capogna, Luca; Danielli, Donatella; Pauls, Scott D.; Tyson, Jeremy

Hardcover

ISBN 10: 3764381329 ISBN 13: 9783764381325

Publisher: Birkhäuser, 2007

Synopsis

This book gives an up-to-date account of progress on Pansu's celebrated problem on the sub-Riemannian isoperimetric profile of the Heisenberg group. It also serves as an introduction to the general field of sub-Riemannian geometric analysis. It develops the methods and tools of sub-Riemannian differential geometry, nonsmooth analysis, and geometric measure theory suitable for attacks on Pansu's problem.

About this title

From the Back Cover:

The past decade has witnessed a dramatic and widespread expansion of interest and activity in sub-Riemannian (Carnot-Caratheodory) geometry, motivated both internally by its role as a basic model in the modern theory of analysis on metric spaces, and externally through the continuous development of applications (both classical and emerging) in areas such as control theory, robotic path planning, neurobiology and digital image reconstruction. The quintessential example of a sub Riemannian structure is the Heisenberg group, which is a nexus for all of the aforementioned applications as well as a point of contact between CR geometry, Gromov hyperbolic geometry of complex hyperbolic space, subelliptic PDE, jet spaces, and quantum mechanics. This book provides an introduction to the basics of sub-Riemannian differential geometry and geometric analysis in the Heisenberg group, focusing primarily on the current state of knowledge regarding Pierre Pansu's celebrated 1982 conjecture regarding the sub-Riemannian isoperimetric profile. It presents a detailed description of Heisenberg submanifold geometry and geometric measure theory, which provides an opportunity to collect for the first time in one location the various known partial results and methods of attack on Pansu's problem. As such it serves simultaneously as an introduction to the area for graduate students and beginning researchers, and as a research monograph focused on the isoperimetric problem suitable for experts in the area.

Review:

Aus den Rezensionen:

 “Es geht um differentialgeometrische Fragestellungen im Falle von Heisenberggruppen und verwandten Räumen ... liefert dies Modelle für eingeschränkte Bewegungsmöglichkeiten ... gibt eine systematische Einführung zu den mathematischen Aspekten der Theorie, beschreibt einige Anwendungen und führt weiter zu aktuellen Forschungsthemen ... Eine umfangreiche Bibliographie liefert weitere Informationen.“ (V. LOSERT, in: Monatshefte für Mathematik, January/2010, Vol. 159, Issue 1-2, S. 212)

About this edition

• Publisher: Birkhäuser
• Publication date: 2007
• ISBN 10: 3764381329
• ISBN 13: 9783764381325
• Binding: Hardcover
• Number of pages: 240