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

235 Pages·2007·1.687 MB·English

by  Luca Capogna, Scott D. Pauls, Donatella Danielli (auth.), Jeremy T. Tyson (eds.)

Progress in Mathematics
Volume 259
Series Editors
H. Bass
J. Oesterlé
A. Weinstein
Luca Capogna
Donatella Danielli
Scott D. Pauls
Jeremy T. Tyson
An Introduction to the
Heisenberg Group
and the Sub-Riemannian
Isoperimetric Problem
Birkhäuser
Basel (cid:120)(cid:3)Boston (cid:120)(cid:3)Berlin(cid:3)
Authors:
Luca Capogna Donatella Danielli
Department of Mathematics Department of Mathematics
University of Arkansas Purdue University
Fayetteville, AR 72701 West Lafayette, IN 47907-1395
USA USA
e-mail: [email protected] e-mail: [email protected]
Scott D. Pauls Jeremy T. Tyson
Department of Mathematics Department of Mathematics
6188 Kemeny Hall University of Illinois at Urbana-Champaign
Dartmouth College 1409 West Green Street
Hanover, NH 03755 Urbana, IL 61801
USA USA
e-mail: [email protected] e-mail: [email protected]
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ISBN 978-3-7643-8132-5 Birkhäuser Verlag AG, Basel - Boston - Berlin
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ISBN-10: 3-7643-8132-9 e-ISBN-10: 3-7643-8133-7
ISBN-13: 978-3-7643-8132-5 e-ISBN-13: 978-3-7643-8133-2
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Dedicated to Nicola Garofalo
on the occasion of his 50th birthday
...mercatique solum,
facti de nomine Byrsam,
taurino quantum possent
circumdare tergo.
(Virgil, Eneid,
Book I, 367–368)
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
1 The Isoperimetric Problem in Euclidean Space
1.1 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 The Heisenberg Group and Sub-Riemannian Geometry
2.1 The first Heisenberg group H . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 The horizontal distribution in H . . . . . . . . . . . . . . . 14
2.1.2 Higher-dimensional Heisenberg groups Hn . . . . . . . . . . 15
2.1.3 Carnot groups . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2 Carnot–Carath´eodorydistance . . . . . . . . . . . . . . . . . . . . 16
2.2.1 Constrained dynamics . . . . . . . . . . . . . . . . . . . . . 16
2.2.2 Sub-Riemannian structure . . . . . . . . . . . . . . . . . . . 19
2.2.3 Carnot groups . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.3 Geodesics and bubble sets . . . . . . . . . . . . . . . . . . . . . . . 22
2.4 Riemannian approximants to the Heisenberg group . . . . . . . . . 24
2.4.1 The g metrics . . . . . . . . . . . . . . . . . . . . . . . . . 25
L
2.4.2 Levi-Civita connection and curvature. . . . . . . . . . . . . 26
2.4.3 Gromov–Hausdorffconvergence . . . . . . . . . . . . . . . . 28
2.4.4 Carnot–Carath´eodorygeodesics . . . . . . . . . . . . . . . . 30
2.4.5 Riemannian approximants to Hn and Carnot groups . . . . 33
2.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Applications of Heisenberg Geometry
3.1 Jet spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2 Applied models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.1 Nonholonomic path planning . . . . . . . . . . . . . . . . . 42
3.2.2 Geometry of the visual cortex . . . . . . . . . . . . . . . . . 43
3.3 CR structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
viii Contents
3.4 Boundary of complex hyperbolic space . . . . . . . . . . . . . . . . 48
3.4.1 Gromov hyperbolic spaces . . . . . . . . . . . . . . . . . . . 48
3.4.2 Gromov boundary and visual metric . . . . . . . . . . . . . 48
3.4.3 Complex hyperbolic space H2 and
C
its boundary at infinity . . . . . . . . . . . . . . . . . . . . 50
3.4.4 The Bergman metric . . . . . . . . . . . . . . . . . . . . . . 51
3.4.5 Boundary at infinity of H2 and the Heisenberg group . . . 53
C
3.5 Further results: geodesics in the roto-translationspace . . . . . . . 55
3.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
4 Horizontal Geometry of Submanifolds
4.1 Invariance of the Sub-Riemannian Metric with
respect to Riemannian extensions . . . . . . . . . . . . . . . . . . . 64
4.2 The second fundamental form in (R3,g ) . . . . . . . . . . . . . . 65
L
4.3 Horizontal geometry of hypersurfaces in H . . . . . . . . . . . . . . 69
4.3.1 Horizontal geometry in Hn . . . . . . . . . . . . . . . . . . 72
4.3.2 Legendrian foliations . . . . . . . . . . . . . . . . . . . . . . 75
4.4 Analysis at the characteristic set and fine regularity of surfaces . . 77
4.4.1 The Legendrian foliation near non-isolated points
of the characteristic locus . . . . . . . . . . . . . . . . . . . 79
4.4.2 The Legendrian foliation near isolated points
of the characteristic locus . . . . . . . . . . . . . . . . . . . 84
4.5 Further results: intrinsically regular surfaces and
the Rumin complex. . . . . . . . . . . . . . . . . . . . . . . . . . . 89
4.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
5 Sobolev and BV Spaces
5.1 Sobolev spaces, perimeter measure and total variation . . . . . . . 95
5.1.1 Riemannian perimeter approximation . . . . . . . . . . . . 98
5.2 A sub-Riemannian Green’s formula and the
fundamental solution of the Heisenberg Laplacian . . . . . . . . . . 100
5.3 Embedding theorems for the Sobolev and BV spaces . . . . . . . . 101
5.3.1 The geometric case
(Sobolev–Gagliardo–Nirenberginequality) . . . . . . . . . . 102
5.3.2 The subcritical case . . . . . . . . . . . . . . . . . . . . . . 105
5.3.3 The supercritical case . . . . . . . . . . . . . . . . . . . . . 106
5.3.4 Compactness of the embedding BV ⊂L1
on John domains . . . . . . . . . . . . . . . . . . . . . . . . 107
5.4 Further results: Sobolev embedding theorems . . . . . . . . . . . . 109
5.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Contents ix
6 Geometric Measure Theory and Geometric Function Theory
6.1 Area and co-area formulas . . . . . . . . . . . . . . . . . . . . . . . 117
6.2 Pansu–Rademachertheorem . . . . . . . . . . . . . . . . . . . . . . 123
6.3 Equivalence of perimeter and Minkowski content . . . . . . . . . . 126
6.4 First variation of the perimeter . . . . . . . . . . . . . . . . . . . . 127
6.4.1 Parametric surfaces and noncharacteristic variations . . . . 128
6.4.2 General variations . . . . . . . . . . . . . . . . . . . . . . . 133
6.5 Mostow’s rigidity theorem for H2 . . . . . . . . . . . . . . . . . . . 135
C
6.5.1 Quasiconformalmappings on H . . . . . . . . . . . . . . . . 139
6.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7 The Isoperimetric Inequality in H
7.1 Equivalence of the isoperimetric and geometric
Sobolev inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . 143
7.2 Isoperimetric inequalities in Hadamard manifolds . . . . . . . . . . 144
7.3 Pansu’s proof of the isoperimetric inequality in H . . . . . . . . . . 147
7.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
8 The Isoperimetric Profile of H
8.1 Pansu’s conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
8.2 Existence of minimizers . . . . . . . . . . . . . . . . . . . . . . . . 154
8.3 Isoperimetric profile has constant mean curvature . . . . . . . . . . 157
8.3.1 Parametrizationof C2 CMC t-graphs in H . . . . . . . . . 159
8.4 Minimizers with symmetries . . . . . . . . . . . . . . . . . . . . . . 162
8.5 The C2 isoperimetric profile in H . . . . . . . . . . . . . . . . . . . 168
8.6 The convex isoperimetric profile of H . . . . . . . . . . . . . . . . . 172
8.7 Other approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
8.7.1 Riemannian approximation approach . . . . . . . . . . . . . 176
8.7.2 Failure of the Brunn–Minkowski approach
to isoperimetry in H . . . . . . . . . . . . . . . . . . . . . . 180
8.7.3 Horizontal mean curvature flow . . . . . . . . . . . . . . . . 181
8.8 Further results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
8.8.1 The isoperimetric problem in the Grushin plane . . . . . . 183
8.8.2 The classification of symmetric CMC surfaces in Hn . . . . 185
8.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186