Geometry, Analysis and Dynamics on Sub-riemannian Manifolds

334 Pages·2016·1.647 MB·English

by  Davide Barilari, Ugo Boscain, Mario Sigalotti

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ISBN 978-3-03719-162-0
www.ems-ph.org
Barilari et al. Vol. I | Rotis Sans | Pantone 287, Pantone 116 | 170 x 240 mm | RB: 16 ?? mm

EMS Series of Lectures in Mathematics
Edited by Ari Laptev (Imperial College, London, UK)
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Alberto Cavicchioli, Friedrich Hegenbarth and Dušan Repovš, Higher-Dimensional Generalized
Manifolds: Surgery and Constructions
Geometry, Analysis
and Dynamics
on sub-Riemannian
Manifolds
Volume I
Davide Barilari
Ugo Boscain
Mario Sigalotti
Editors
Editors:
Prof. Davide Barilari Prof. Mario Sigalotti
Institut de Mathématiques de Jussieu-Paris Rive Gauche INRIA Saclay
Université Paris 7, Denis Diderot Centre de Mathématiques Appliquées
5 rue Thomas Mann École Polytechnique
75205 Paris 13 Cedex Route de Saclay
France 91128 Palaiseau Cedex
France
E-mail: [email protected]
E-mail: [email protected]
Prof. Ugo Boscain
CNRS
Centre de Mathématiques Appliquées, Ecole Polytechnique
Route de Saclay
91128 Palaiseau Cedex
France
E-mail: [email protected]
2010 Mathematics Subject Classification: Primary: 53C17; Secondary: 35H10, 60H30, 49J15
Key words: sub-Riemannian geometry, hypoelliptic operators, non-holonomic constraints, optimal control,
rough paths
ISBN 978-3-03719-162-0
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9 8 7 6 5 4 3 2 1
Preface
This book, divided into two volumes, collects different cycles of lectures given at
the IHP Trimester “Geometry, Analysis and Dynamics on Sub-Riemannian Mani-
folds”,heldatInstitutHenriPoincaréinParis,andtheCIRMSummerSchool“Sub-
Riemannian Manifolds: From Geodesics to Hypoelliptic Diffusion”, held at Centre
InternationaledeRencontresMathématiques,inLuminy,duringfall2014.
Sub-Riemannian geometry is a generalization of Riemannian geometry, whose
birthdatesbacktoCarathéodory’s1909seminalpaperonthefoundationsofCarnot
thermodynamics,followedbyE.Cartan’s1928addressattheInternationalCongress
ofMathematiciansinBologna.
Sub-Riemannian geometry is characterized by non-holonomic constraints: dis-
tances are computed by minimizing the length of curves whose velocities belong
to a given subspace of the tangent space. From the theoretical point of view, sub-
Riemannian geometry is the geometry underlying the theory of hypoelliptic opera-
torsanddegeneratediffusionsonmanifolds.
In the last twenty years, sub-Riemannian geometry has emerged as an indepen-
dent research domain, with extremely rich motivations and ramifications in several
partsofpureandappliedmathematics. Letusmentiongeometricanalysis,geomet-
ricmeasuretheory, stochasticcalculusandevolutionequationstogetherwithappli-
cationsinmechanicsandoptimalcontrol(motionplanning,robotics,nonholonomic
mechanics,quantumcontrol)andanothertoimageprocessing,biologyandvision.
Evenif,nowadays,sub-Riemanniangeometryisrecognizedasatransversesub-
ject,researchersworkingindifferentcommunitiesarestillusingquitedifferentlan-
guage. Theaimoftheselecturesistocollectreferencematerialonsub-Riemannian
structuresfortheuseofbothresearchersandgraduatestudents. Startingfrombasic
definitions and extending up to the frontiers of research, this material reflects the
pointofviewofauthorswithdifferentbackgrounds. Theexchangesamongthepar-
ticipantsoftheIHPTrimesterandoftheCIRMschoolarereflectedherebyseveral
connections and interplays between the different chapters. This will hopefully re-
ducetheexistinggapinlanguagebetweenthedifferentcommunitiesandfavourthe
futuredevelopmentofthefield.
ThenotesofFrancescoSerraCassanogiveanextensivepresentationofgeomet-
ric measure theory in Carnot groups. The first part of the notes discusses differen-
tialcalculusformapsbetweenCarnotgroupsinrelationwiththeunderlyingmetric
structure. The text then focuses on differential calculus within Carnot groups and
uses it to investigate intrinsic regular and Lipschitz surfaces in Carnot groups and
their relation with rectifiability. The final section deals with sets of finite perimeter
and with the related notions of reduced and minimal boundary. An application to
minimalgraphsinHeisenberggroupsisdeveloped.
vi Preface
The lecture notes by Nicola Garofalo are a quite comprehensive compendium
of results in geometric analysis. In the first part, starting from basic examples and
definitions of sub-Riemannian manifolds, length-spaces and Carnot groups, he dis-
cusses, in the sub-Riemannian context, Sobolev spaces, BV functions and Sobolev
embeddingtheorems,passingthroughisoperimetricinequalities. Inthesecondpart
hediscussesclassicalresultsingeometricanalysisinRiemannianmanifoldsandthe
now classical contributions by Folland–Stein, Rothschild–Stein and Nagel–Stein–
Wainger in the sub-Riemannian context. Besides giving estimates for the funda-
mental solution of the heat equation, the goal is to discuss Li–Yau inequalities and
curvaturedimensionalinequalitiesinthesub-Riemanniancase.
ThelecturenotesbyFabriceBaudoinstudyhypoellipticdiffusionoperatorsfrom
the viewpoint of geometric analysis. The main focus is on sub-Riemannian Lapla-
cians that arise as horizontal Laplacians of a Riemannian foliation. For this kind
of operator an extensive theory is developed, with special attention to subelliptic
Weitzenböckidentitiesanddifferentapplications,fromLi–Yauinequalitiestospec-
tral gap inequalities and the Bonnet–Myers theorem. The last section is devoted to
theanalysisofsomeKolmogorov-typehypoellipticdiffusionoperatorsandhypoco-
erciveestimates.
DavideBarilari
UgoBoscain
MarioSigalotti
Contents
1 SometopicsofgeometricmeasuretheoryinCarnotgroups. . . . . . . . . 1
FrancescoSerraCassano
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 AnintroductiontoCarnotgroups . . . . . . . . . . . . . . . . . . . 3
3 DifferentialcalculusonCarnotgroups . . . . . . . . . . . . . . . . 20
4 DifferentialcalculuswithinCarnotgroups . . . . . . . . . . . . . . 34
5 SetsoffiniteperimeterandminimalsurfacesinCarnotgroups . . . 83
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
2 Hypoelliptic operators and some aspects of analysis and geometry of sub-
Riemannianspaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
NicolaGarofalo
1 Sub-Riemanniangeometryandhypoellipticoperators . . . . . . . . 123
2 Carnotgroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
3 FundamentalsolutionsandtheYamabeequation. . . . . . . . . . . 147
4 Carnot–Carathéodorydistance . . . . . . . . . . . . . . . . . . . . 160
5 SobolevandBVspaces . . . . . . . . . . . . . . . . . . . . . . . . 175
6 Fractionalintegrationinspacesofhomogeneoustype . . . . . . . . 189
7 Fundamentalsolutionsofhypoellipticoperators . . . . . . . . . . . 202
8 ThegeometricSobolevembeddingandtheisoperimetricinequality. 212
9 TheLi–YauinequalityforcompletemanifoldswithRicci≥0. . . . 216
10 HeatsemigroupapproachtotheLi–Yauinequality . . . . . . . . . 224
11 Aheatequationapproachtothevolumedoublingproperty . . . . . 233
12 Asub-Riemanniancurvature-dimensioninequality . . . . . . . . . 239
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
3 Sub-Laplacians and hypoelliptic operators on totally geodesic Riemannian
foliations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
FabriceBaudoin
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
2 RiemannianfoliationsandtheirLaplacians . . . . . . . . . . . . . 261
3 HorizontalLaplaciansandheatkernelsonmodelspaces . . . . . . 268
4 TransverseWeitzenböckformulas . . . . . . . . . . . . . . . . . . 282
5 Thehorizontalheatsemigroup . . . . . . . . . . . . . . . . . . . . 292
6 ThehorizontalBonnet–Myerstheorem . . . . . . . . . . . . . . . . 302
7 Riemannianfoliationsandhypocoercivity . . . . . . . . . . . . . . 308
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323

Chapter 1
Some topics of geometric measure theory in
Carnot groups
FrancescoSerraCassano(cid:49)
Tomyparents
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 AnintroductiontoCarnotgroups. . . . . . . . . . . . . . . . . . . . . . . . . 3
3 DifferentialcalculusonCarnotgroups. . . . . . . . . . . . . . . . . . . . . . 20
4 DifferentialcalculuswithinCarnotgroups. . . . . . . . . . . . . . . . . . . . 34
5 SetsoffiniteperimeterandminimalsurfacesinCarnotgroups. . . . . . . . . . 83
Bibliography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
1 Introduction
These notes aim at illustrating some results achieved in geometric measure theory
in Carnot groups. They are an extended version of part of the course Geomet-
ric Measure Theory given during the “Geometry, Analysis and Dynamics on sub-
Riemannian Manifolds” trimester, held in Paris in September 2014 at the Institut
HenriPoincaré.
Firstofall,IwouldliketothanktheorganizersofthetrimesterAndreiAgrachev,
Davide Barilari, Ugo Boscain, Yacine Chitour, Frederic Jean, Ludovic Rifford, and
MarioSigalotti,fortheirkindinvitation,aswelltoIHPforitsbacking.
Itisalsoagreatpleasureformetoacknowledgethehelpandsupportofseveral
friends of mine who have made this work possible: first of all, most of the results
(cid:49)[email protected]
DipartimentodiMatematica,UniversitàdiTrento,ViaSommarive14,38123,Trento,Italy.
F.S.C.issupportedbyMIUR,Italy,GNAMPAoftheINdAM,UniversityofTrento,ItalyandbyMAnETMarie
CurieInitialTrainingNetworkGrant607643–FP7-PEOPLE-2013-ITN.PartoftheworkwasdonewhileF.S.C.
wasvisitingattheInstitutHenriPoincaré,Paris.HewishestothanktheIHPforitshospitality.