An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem [electronic resource] / by Luca Capogna, Scott D. Pauls, Donatella Danielli ; edited by Jeremy T. Tyson.
Contributor(s): Pauls, Scott D [author.] | Danielli, Donatella [author.] | Tyson, Jeremy T [editor.] | SpringerLink (Online service)Material type: TextSeries: Progress in Mathematics: 259Publisher: Basel : Birkhäuser Basel, 2007Description: XVI, 224 p. online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9783764381332Subject(s): Mathematics | Topological groups | Lie groups | Global analysis (Mathematics) | Manifolds (Mathematics) | Partial differential equations | System theory | Differential geometry | Complex manifolds | Mathematics | Differential Geometry | Topological Groups, Lie Groups | Manifolds and Cell Complexes (incl. Diff.Topology) | Partial Differential Equations | Global Analysis and Analysis on Manifolds | Systems Theory, ControlAdditional physical formats: Printed edition:: No titleDDC classification: 516.36 LOC classification: QA641-670Online resources: Click here to access online
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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.
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.