Capogna / Danielli / Tyson An Introduction to the Heisenberg Group and the Sub-Riemannian Isoperimetric Problem

1. Auflage 2007
ISBN: 978-3-7643-8133-2
Verlag: Springer
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 224 Seiten

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.

Autoren/Hrsg.

Capogna, Luca

Danielli, Donatella

Tyson, Jeremy

Inhaltsverzeichnis

1;Contents;8
2;Preface;12
3;The Isoperimetric Problem in Euclidean Space;18
3.1;1.1 Notes;25
4;The Heisenberg Group and Sub- Riemannian Geometry;27
4.1;2.1 The first Heisenberg group;27
4.2;2.2 Carnot–Carath ´ eodory distance;32
4.3;2.3 Geodesics and bubble sets;38
4.4;2.4 Riemannian approximants to;40
4.5;2.5 Notes;50
5;Applications of Heisenberg Geometry;54
5.1;3.1 Jet spaces;54
5.2;3.2 Applied models;55
5.3;3.3 CR structures;60
5.4;3.4 Boundary of complex hyperbolic space;63
5.5;3.5 Further results: geodesics in the roto- translation space;70
5.6;3.6 Notes;73
6;Horizontal Geometry of Submanifolds;77
6.1;4.1 Invariance of the Sub-Riemannian Metric with respect to Riemannian extensions;78
6.2;4.2 The second fundamental form in;79
6.3;4.3 Horizontal geometry of hypersurfaces in;83
6.4;4.4 Analysis at the characteristic set and fine regularity of surfaces;91
6.5;4.5 Further results: intrinsically regular surfaces and the Rumin complex;103
6.6;4.6 Notes;105
7;Sobolev and BV Spaces;108
7.1;5.1 Sobolev spaces, perimeter measure and total variation;108
7.2;5.2 A sub-Riemannian Green’s formula and the fundamental solution of the Heisenberg Laplacian;113
7.3;5.3 Embedding theorems for the Sobolev and BV spaces;114
7.4;5.4 Further results: Sobolev and Sobolev–Poincar ´ e embedding theorems and analysis in metric spaces;122
7.5;5.5 Notes;125
8;Geometric Measure Theory and Geometric Function Theory;129
8.1;6.1 Area and co-area formulas;129
8.2;6.2 Pansu–Rademacher theorem;135
8.3;6.3 Equivalence of perimeter and Minkowski content;138
8.4;6.4 First variation of the perimeter;139
8.5;6.5 Mostow’s rigidity theorem for;147
8.6;6.6 Notes;152
9;The Isoperimetric Inequality in H;155
9.1;7.1 Equivalence of the isoperimetric and geometric Sobolev inequalities;155
9.2;7.2 Isoperimetric inequalities in Hadamard manifolds;156
9.3;7.3 Pansu’s proof of the isoperimetric inequality in;159
9.4;7.4 Notes;162
10;The Isoperimetric Profile of;163
10.1;8.1 Pansu’s conjecture;163
10.2;8.2 Existence of minimizers;166
10.3;8.3 Smooth isoperimetric profiles have constant horizontal mean curvature;169
10.4;8.4 Existence and characterization of minimizers with additional symmetries;174
10.5;8.5 The C2 isoperimetric profile in H;180
10.6;8.6 The convex isoperimetric profile of;184
10.7;8.7 Other approaches;188
10.8;8.8 Further results;195
10.9;8.9 Notes;198
11;Best Constants for Other Geometric Inequalities on the Heisenberg Group;203
11.1;9.1 L2-Sobolev embedding theorem;203
11.2;9.2 Moser–Trudinger inequality;207
11.3;9.3 Hardy inequality;211
11.4;9.4 Notes;212
12;Bibliography;215
13;Index;231