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

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

• ISBN13: 9783764381325
• ISBN10: 3764381329
• Format: Hardcover
• Copyright: 2007-07-04
• Publisher: Birkhauser

Summary

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

Preface	p. xi
The Isoperimetric Problem in Euclidean Space
Notes	p. 8
The Heisenberg Group and Sub-Riemannian Geometry
The first Heisenberg group <$>{\op H}<$>	p. 11
The horizontal distribution in <$>{\op H}<$>	p. 14
Higher-dimensional Heisenberg groups <$>{\op H}^n<$>	p. 15
Carnot groups	p. 15
Carnot-Carathéodory distance	p. 16
Constrained dynamics	p. 16
Sub-Riemannian structure	p. 19
Carnot groups	p. 21
Geodesies and bubble sets	p. 22
Riemannian approximants to the Heisenberg group	p. 24
The gL metrics	p. 25
Levi-Civita connection and curvature	p. 26
Gromov-Hausdorff convergence	p. 28
Carnot-Carathéodory geodesics	p. 30
Riemannian approximants to <$>{\op H}^n<$> and Carnot groups	p. 33
Notes	p. 34
Applications of Heisenberg Geometry
Jet spaces	p. 39
Applied models	p. 40
Nonholonomic path planning	p. 42
Geometry of the visual cortex	p. 43
CR structures	p. 45
Boundary of complex hyperbolic space	p. 48
Gromov hyperbolic spaces	p. 48
Gromov boundary and visual metric	p. 48
Complex hyperbolic space <$>H_{{\op C}}^2<$> and its boundary at infinity	p. 50
The Bergman metric	p. 51
Boundary at infinity of <$>H_{{\op C}}^2<$> and the Heisenberg group	p. 53
Further results: geodesics in the roto-translation space	p. 55
Notes	p. 58
Horizontal Geometry of Submanifolds
Invariance of the Sub-Riemannian Metric with respect to Riemannian extensions	p. 64
The second fundamental form in <$>({\op R}^3,g_L)<$>	p. 65
Horizontal geometry of hypersurfaces in <$>{\op H}<$>	p. 69
Horizontal geometry in <$>{\op H}^n<$>	p. 72
Legendrian foliations	p. 75
Analysis at the characteristic set and fine regularity of surfaces	p. 77
The Legendrian foliation near non-isolated points of the characteristic locus	p. 79
The Legendrian foliation near isolated points of the characteristic locus	p. 84
Further results: intrinsically regular surfaces and the Rumin complex	p. 89
Notes	p. 91
Sobolev and BV Spaces
Sobolev spaces, perimeter measure and total variation	p. 95
Riemannian perimeter approximation	p. 98
A sub-Riemannian Green's formula and the fundamental solution of the Heisenberg Laplacian	p. 100
Embedding theorems for the Sobolev and BV spaces	p. 101
The geometric case (Sobolev-Gagliardo-Nirenberg inequality)	p. 102
The subcritical case	p. 105
The supercritical case	p. 106
Compactness of the embedding BV ⊂ L1 on John domains	p. 107
Further results: Sobolev embedding theorems	p. 109
Notes	p. 112
Geometric Measure Theory and Geometric Function Theory
Area and co-area formulas	p. 117
Pansu-Rademacher theorem	p. 123
Equivalence of perimeter and Minkowski content	p. 126
First variation of the perimeter	p. 127
Parametric surfaces and noncharacteristic variations	p. 128
General variations	p. 133
Mostow's rigidity theorem for <$>H_{{\op C}}^2<$>	p. 135
Quasiconformal mappings on <$>{\op H}<$>	p. 139
Notes	p. 140
The Isoperimetric Inequality in <$>{\op H}<$>
Equivalence of the isoperimetric and geometric Sobolev inequalities	p. 143
Isoperimetric inequalities in Hadamard manifolds	p. 144
Pansu's proof of the isoperimetric inequality in <$>{\op H}<$>	p. 147
Notes	p. 150
The Isoperimetric Profile of <$>{\op H}<$>
Pansu's conjecture	p. 151
Existence of minimizers	p. 154
Isoperimetric profile has constant mean curvature	p. 157
Parametrization of C2 CMC t-graphs in <$>{\op H}<$>	p. 159
Minimizers with symmetries	p. 162
The C2 isoperimetric profile in <$>{\op H}<$>	p. 168
The convex isoperimetric profile of <$>{\op H}<$>	p. 172
Other approaches	p. 176
Riemannian approximation approach	p. 176
Failure of the Brunn-Minkowski approach to isoperimetry in <$>{\op H}<$>	p. 180
Horizontal mean curvature flow	p. 181
Further results	p. 183
The isoperimetric problem in the Grushin plane	p. 183
The classification of symmetric CMC surfaces in <$>{\op H}^n<$>	p. 185
Notes	p. 186
Best Constants for Other Geometric Inequalities on the Heisenberg Group
L2-Sobolev embedding theorem	p. 191
Moser-Trudinger inequality	p. 195
Hardy inequality	p. 199
Notes	p. 200
Bibliography	p. 203
Index	p. 219
List of Figures
Dido, Queen of Carthage. Engraving by Mathäus Merian the Elder, 1630	p. 1
Isoperimetric sets in <$>{\op R}^2<$> have symmetry	p. 6
Isoperimetric sets are convex	p. 6
Examples of horizontal planes at different points	p. 14
Horizontal paths connecting points in <$>{\op H}<$>	p. 17
The conjectured isoperimetric set in <$>{\op H}^1<$>	p. 24
Coordinates describing the unicycle	p. 43
The hypercolumn structure of V1	p. 44
Illustration of Pansu's approach	p. 145
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