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

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...

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Main Authors: Capogna, Luca, Danielli, Donatella (Author), Pauls, Scott D. (Author), Tyson, Jeremy (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: Basel Birkhäuser Basel 2007, 2007
Edition:1st ed. 2007
Series:Progress in Mathematics
Subjects:
Online Access:
Collection: Springer eBooks 2005- - Collection details see MPG.ReNa
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245 0 0 |a An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem  |h Elektronische Ressource  |c by Luca Capogna, Donatella Danielli, Scott D. Pauls, Jeremy Tyson 
250 |a 1st ed. 2007 
260 |a Basel  |b Birkhäuser Basel  |c 2007, 2007 
300 |a XVI, 224 p  |b online resource 
505 0 |a The Isoperimetric Problem in Euclidean Space -- The Heisenberg Group and Sub-Riemannian Geometry -- Applications of Heisenberg Geometry -- Horizontal Geometry of Submanifolds -- Sobolev and BV Spaces -- Geometric Measure Theory and Geometric Function Theory -- The Isoperimetric Inequality in ? -- The Isoperimetric Profile of ? -- Best Constants for Other Geometric Inequalities on the Heisenberg Group 
653 |a Differential Geometry 
653 |a Differential equations, partial 
653 |a Global Analysis and Analysis on Manifolds 
653 |a Systems theory 
653 |a Global differential geometry 
653 |a Cell aggregation / Mathematics 
653 |a Partial Differential Equations 
653 |a Topological Groups 
653 |a Systems Theory, Control 
653 |a Manifolds and Cell Complexes (incl. Diff.Topology) 
653 |a Global analysis 
653 |a Topological Groups, Lie Groups 
700 1 |a Danielli, Donatella  |e [author] 
700 1 |a Pauls, Scott D.  |e [author] 
700 1 |a Tyson, Jeremy  |e [author] 
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989 |b Springer  |a Springer eBooks 2005- 
490 0 |a Progress in Mathematics 
856 |u https://doi.org/10.1007/978-3-7643-8133-2?nosfx=y  |x Verlag  |3 Volltext 
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520 |a 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