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

Luca Capogna, Donatella Danielli, Scott D. Pauls, Jeremy Tyson (Autoren)

eBook Download: PDF

2007 | 2007
XVI, 224 Seiten
Springer Basel (Verlag)
978-3-7643-8133-2 (ISBN)

Infos

This book gives an up-to-date account of progress on Pansu's celebrated problem on the sub-Riemannian isoperimetric profile of the Heisenberg group. It also serves as an introduction to the general field of sub-Riemannian geometric analysis. It develops the methods and tools of sub-Riemannian differential geometry, nonsmooth analysis, and geometric measure theory suitable for attacks on Pansu's problem.

Inhalt

Contents 8
Preface 12
The Isoperimetric Problem in Euclidean Space 18
1.1 Notes 25
The Heisenberg Group and Sub- Riemannian Geometry 27
2.1 The first Heisenberg group 27
2.2 Carnot–Carath ´ eodory distance 32
2.3 Geodesics and bubble sets 38
2.4 Riemannian approximants to 40
2.5 Notes 50
Applications of Heisenberg Geometry 54
3.1 Jet spaces 54
3.2 Applied models 55
3.3 CR structures 60
3.4 Boundary of complex hyperbolic space 63
3.5 Further results: geodesics in the roto- translation space 70
3.6 Notes 73
Horizontal Geometry of Submanifolds 77
4.1 Invariance of the Sub-Riemannian Metric with respect to Riemannian extensions 78
4.2 The second fundamental form in 79
4.3 Horizontal geometry of hypersurfaces in 83
4.4 Analysis at the characteristic set and fine regularity of surfaces 91
4.5 Further results: intrinsically regular surfaces and the Rumin complex 103
4.6 Notes 105
Sobolev and BV Spaces 108
5.1 Sobolev spaces, perimeter measure and total variation 108
5.2 A sub-Riemannian Green’s formula and the fundamental solution of the Heisenberg Laplacian 113
5.3 Embedding theorems for the Sobolev and BV spaces 114
5.4 Further results: Sobolev and Sobolev–Poincar ´ e embedding theorems and analysis in metric spaces 122
5.5 Notes 125
Geometric Measure Theory and Geometric Function Theory 129
6.1 Area and co-area formulas 129
6.2 Pansu–Rademacher theorem 135
6.3 Equivalence of perimeter and Minkowski content 138
6.4 First variation of the perimeter 139
6.5 Mostow’s rigidity theorem for 147
6.6 Notes 152
The Isoperimetric Inequality in H 155
7.1 Equivalence of the isoperimetric and geometric Sobolev inequalities 155
7.2 Isoperimetric inequalities in Hadamard manifolds 156
7.3 Pansu’s proof of the isoperimetric inequality in 159
7.4 Notes 162
The Isoperimetric Profile of 163
8.1 Pansu’s conjecture 163
8.2 Existence of minimizers 166
8.3 Smooth isoperimetric profiles have constant horizontal mean curvature 169
8.4 Existence and characterization of minimizers with additional symmetries 174
8.5 The C2 isoperimetric profile in H 180
8.6 The convex isoperimetric profile of 184
8.7 Other approaches 188
8.8 Further results 195
8.9 Notes 198
Best Constants for Other Geometric Inequalities on the Heisenberg Group 203
9.1 L2-Sobolev embedding theorem 203
9.2 Moser–Trudinger inequality 207
9.3 Hardy inequality 211
9.4 Notes 212
Bibliography 215
Index 231

Produkt-Details

• Erscheint lt. Verlag: 8.8.2007
• Reihe/Serie: Progress in Mathematics
• Reihe/Serie: Progress in Mathematics
• Zusatzinfo: XVI, 224 p.
• Verlagsort: Basel
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik
• Schlagworte: Cauchy-Riemann manifold • Contact Geometry • Differential Geometry • Evolution Equations • geometric measure theory • immersions • manifold • minimal surface • Partial differential equations • quasiconformal mapping • Riemannian Geometry • Sobolev Space • sub-Riemannian geometry
• ISBN-10: 3-7643-8133-7 / 3764381337
• ISBN-13: 978-3-7643-8133-2 / 9783764381332