Geometry, Analysis and Dynamics on Sub-riemannian Manifolds PDF
Preview Geometry, Analysis and Dynamics on Sub-riemannian Manifolds
Series of Lectures in Mathematics Geometry, Analysis and Dynamics sG u e bo on sub-Riemannian Manifolds -m R ie et mr y Volume I a, Geometry, Analysis nA nn ia a l ny Davide Barilari, Ugo Boscain and Mario Sigalotti s and Dynamics Mi s Editors a a nn ifd on sub-Riemannian o D l dy sn Sub-Riemannian manifolds model media with constrained dynamics: motion at any , a point is only allowed along a limited set of directions, which are prescribed by the Vm Manifolds o physical problem. From the theoretical point of view, sub-Riemannian geometry is the luic ms geometry underlying the theory of hypoelliptic operators and degenerate diffusions on o e manifolds. n I Volume I In the last twenty years, sub-Riemannian geometry has emerged as an independent research domain, with extremely rich motivations and ramifications in several parts of pure and applied mathematics, such as geometric analysis, geometric measure theory, stochastic calculus and evolution equations together with applications in mechanics, MD Davide Barilari optimal control and biology. aa riovid e Ugo Boscain The aim of the lectures collected here is to present sub-Riemannian structures for the Sig Ba use of both researchers and graduate students. ar loila Mario Sigalotti tti, Eri, U dg ito Editors o B rso s c a in a n d 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) EMS Series of Lectures in Mathematics is a book series aimed at students, professional mathematicians and scientists. It publishes polished notes arising from seminars or lecture series in all fields of pure and applied mathematics, including the reissue of classic texts of continuing interest. The individual volumes are intended to give a rapid and accessible introduction into their particular subject, guiding the audience to topics of current research and the more advanced and specialized literature. Previously published in this series: Katrin Wehrheim, Uhlenbeck Compactness Torsten Ekedahl, One Semester of Elliptic Curves Sergey V. Matveev, Lectures on Algebraic Topology Joseph C. Várilly, An Introduction to Noncommutative Geometry Reto Müller, Differential Harnack Inequalities and the Ricci Flow Eustasio del Barrio, Paul Deheuvels and Sara van de Geer, Lectures on Empirical Processes Iskander A. Taimanov, Lectures on Differential Geometry Martin J. Mohlenkamp and María Cristina Pereyra, Wavelets, Their Friends, and What They Can Do for You Stanley E. Payne and Joseph A. Thas, Finite Generalized Quadrangles Masoud Khalkhali, Basic Noncommutative Geometry Helge Holden, Kenneth H. Karlsen, Knut-Andreas Lie and Nils Henrik Risebro, Splitting Methods for Partial Differential Equations with Rough Solutions Koichiro Harada, “Moonshine” of Finite Groups Yurii A. Neretin, Lectures on Gaussian Integral Operators and Classical Groups Damien Calaque and Carlo A. Rossi, Lectures on Duflo Isomorphisms in Lie Algebra and Complex Geometry Claudio Carmeli, Lauren Caston and Rita Fioresi, Mathematical Foundations of Supersymmetry Hans Triebel, Faber Systems and Their Use in Sampling, Discrepancy, Numerical Integration Koen Thas, A Course on Elation Quadrangles Benoît Grébert and Thomas Kappeler, The Defocusing NLS Equation and Its Normal Form Armen Sergeev, Lectures on Universal Teichmüller Space Matthias Aschenbrenner, Stefan Friedl and Henry Wilton, 3-Manifold Groups Hans Triebel, Tempered Homogeneous Function Spaces Kathrin Bringmann, Yann Bugeaud, Titus Hilberdink and Jürgen Sander, Four Faces of Number Theory 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 The Swiss National Library lists this publication in The Swiss Book, the Swiss national bibliography, and the detailed bibliographic data are available on the Internet at http://www.helveticat.ch. This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. © European Mathematical Society 2016 Contact address: European Mathematical Society Publishing House Seminar for Applied Mathematics ETH-Zentrum SEW A27 CH-8092 Zürich Switzerland Phone: +41 (0)44 632 34 36 Email: [email protected] Homepage: www.ems-ph.org Typeset using the authors’ TEX files: Alison Durham, Manchester, UK Printing and binding: Beltz Bad Langensalza GmbH, Bad Langensalza, Germany ∞ Printed on acid free paper 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.
The list of books you might like
