An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem

Luca Capogna, Scott D. Pauls, Donatella Danielli (auth.), Jeremy T. Tyson (eds.)3764381329, 9783764381325, 9783764381332, 3764381337

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.

Table of contents :
Front Matter….Pages i-xvi
The Isoperimetric Problem in Euclidean Space….Pages 1-9
The Heisenberg Group and Sub-Riemannian Geometry….Pages 11-37
Applications of Heisenberg Geometry….Pages 39-61
Horizontal Geometry of Submanifolds….Pages 63-93
Sobolev and BV Spaces….Pages 95-115
Geometric Measure Theory and Geometric Function Theory….Pages 117-142
The Isoperimetric Inequality in ℍ….Pages 143-150
The Isoperimetric Profile of ℍ….Pages 151-190
Best Constants for Other Geometric Inequalities on the Heisenberg Group….Pages 191-202
Back Matter….Pages 203-223