04843nam a22006375i 4500 978-3-7643-8133-2 DE-He213 20210616020215.0 cr nn 008mamaa 100301s2007 sz | s |||| 0|eng d 9783764381332 978-3-7643-8133-2 10.1007/978-3-7643-8133-2 doi QA641-670 PBMP bicssc MAT012030 bisacsh PBMP thema 516.36 23 Capogna, Luca. author. aut http://id.loc.gov/vocabulary/relators/aut An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem [electronic resource] / by Luca Capogna, Donatella Danielli, Scott D. Pauls, Jeremy Tyson. 1st ed. 2007. Basel : Birkhäuser Basel : Imprint: Birkhäuser, 2007. XVI, 224 p. online resource. text txt rdacontent computer c rdamedia online resource cr rdacarrier text file PDF rda Progress in Mathematics, 0743-1643 ; 259 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. Differential geometry. Topological groups. Lie groups. Manifolds (Mathematics). Complex manifolds. Partial differential equations. Global analysis (Mathematics). System theory. Differential Geometry. https://scigraph.springernature.com/ontologies/product-market-codes/M21022 Topological Groups, Lie Groups. https://scigraph.springernature.com/ontologies/product-market-codes/M11132 Manifolds and Cell Complexes (incl. Diff.Topology). https://scigraph.springernature.com/ontologies/product-market-codes/M28027 Partial Differential Equations. https://scigraph.springernature.com/ontologies/product-market-codes/M12155 Global Analysis and Analysis on Manifolds. https://scigraph.springernature.com/ontologies/product-market-codes/M12082 Systems Theory, Control. https://scigraph.springernature.com/ontologies/product-market-codes/M13070 Danielli, Donatella. author. aut http://id.loc.gov/vocabulary/relators/aut Pauls, Scott D. author. aut http://id.loc.gov/vocabulary/relators/aut Tyson, Jeremy. author. aut http://id.loc.gov/vocabulary/relators/aut SpringerLink (Online service) Springer Nature eBook Printed edition: 9783764391850 Printed edition: 9783764381325 Progress in Mathematics, 0743-1643 ; 259 https://doi.org/10.1007/978-3-7643-8133-2 ZDB-2-SMA ZDB-2-SXMS Mathematics and Statistics (SpringerNature-11649) Mathematics and Statistics (R0) (SpringerNature-43713)