Table Of ContentProgress in NonlinearDifferential Equations
and TheirApplications
Volume 53
Editor
HaimBrezis
UniversitePierreetMarieCurie
Paris
and
RutgersUniversity
NewBrunswick,N.J.
EditorialBoard
AntonioAmbrosetti,ScuolaNormaleSuperiore,Pisa
A. Bahri,RutgersUniversity,NewBrunswick
FelixBrowder,RutgersUniversity,NewBrunswick
LuisCafarelli,InstituteforAdvancedStudy,Princeton
LawrenceC.Evans,UniversityofCalifornia,Berkeley
MarianoGiaquinta,UniversityofPisa
DavidKinderlehrer,Carnegie-MellonUniversity,Pittsburgh
SergiuKlainerman,PrincetonUniversity
RobertKohn,NewYorkUniversity
P.L.Lions,UniversityofParisIX
JeanMawhin,UniversiteCatholiquedeLouvain
LouisNirenberg,New YorkUniversity
LambertusPeletier,UniversityofLeiden
PaulRabinowitz,UniversityofWisconsin,Madison
JohnToland,UniversityofBath
Abbas Bahri
Flow Lines and
Algebraic Invariants
in Contact Form Geometry
Springer Science+Business Media, LLC
Abbas Bahri
Rutgers University
Department of Mathematics
New Brunswick, NJ 08903
U.S.A.
Library of Congress Cataloging-in-Publication Data
Bahri, Abbas.
Flow lines and algebraic invariants in contact form geometry / Abbas Bahri.
p. cm. - (Progress in nonlinear differential equations and their applications ; v. 53)
Includes bibliographical references.
ISBN 978-1-4612-6576-4 ISBN 978-1-4612-0021-5 (eBook)
DOI 10.1007/978-1-4612-0021-5
1. Global differential geometry. 2. Riemannian manifolds. 3. Manifolds (Mathematics)
1. Title. II. Series.
QA670.B34 2003
516'.3'62-dc21 2002038316
CIP
AMS Subject Classifications: 53C15, 53C21, 58F05, 70H05, 34A26, 35A15, 55P35
Printed on acid-free paper
© 2003 Springer Science+Business Media New York
Originally published by Birkhauser Boston in 2003
Softcover reprint of the hardcover 1s t edition 2003
AlI rights reserved. This work may not be translated or copied in whole or in part without the writ
ten permission of the publisher (Springer Science+Business Media, LLC), except for brief ex
cerpts in connection with reviews or scholarly analysis. U se in connection with any form of infor
mation storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar
methodology now known or hereafter deve10ped is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication, even if
the former are not especialIy identified, is not to be taken as a sign that such names, as unde stood
by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone.
SPIN 10901762
Reformatted from author's files by John Spiegelman, Abbington, PA.
987 6 5 4 3 2 1
Contents
Prologue 1
Introduction,StatementofResults,andDiscussion
ofRelatedHypotheses 3
1 Topologicalresults 4
2 Intermediatehypotheses(A4), (A4)',(AS),(A6) 7
3 Thenon-Fredholmcharacterofthis variationalproblem, the
associatedcones,condition(AS)(discussionandremoval)....8
4.a Hypothesis(A4)andstatementofthemostgeneralresults,
discussion of(A4) 10
4.b Discussionof(A2),(A3),and(A4) 12
OutlineoftheBook 15
I Review ofthe Previous Results, Some Open Questions..17
I.A SetupoftheVariationalProblem 19
LA.l Introduction 19
LA.2 The variationalframework 20
LA.3 The topology ofCf3 .•.•............•......•..•....•••.•..22
LA.4 Bundlesatinfinity 26
rr2k
LA.4.a andthespaceofv-verticals, thecharacteristic
pieces 26
I.A.4.b The Hd-bundles 26
LA.4.c Neighborhoodsofinfinity 27
LA.S Non-Fredholmbehavior,isotopicdeformationofcurves 28
LA.6 Descriptionofthe variouscriticalpointsatinfinityandtheir
associatedcones 30
LA.6.1 Descriptionofthevariouscriticalpointsatinfinity 30
LA.6.2 Thedifferenceoftopology 32
LA.6.2.a TheMorseindex,differenceoftopology dueto
acriticalpointatinfinityofthefirsttypeandnot
ofthethirdtype 32
vi Contents
LA.6.2.b Differenceoftopologyduetoacriticalpointor
criticalpointatinfinityofthethirdkind.....32
I.A.6.2.c Differenceoftopologyduetoafalsecritical
z-
pointatinfinityofthesecondkind 34
LA.6.2.d Thedifferenceoftopologyduetoacriticalpoint
atinfinityofthesecondandthirdkinds ..... 35
LA.6.2.e Thedifferenceoftopologyduetoacriticalpoint
atinfinityofmixedtypeandthethirdkind...35
I.B TheFlow Zoof[2]: CriticalPointsatInfinity,Falseand
Thue 37
LB.l Abriefdescriptionoftheflow Zodefinedin [2] 37
LB.2 The HJ-flow 39
LB.3 Theflow atinfinity 49
LB.4 Falsecriticalpointsatinfinityofthefirsttype 50
LB.5 Ajustificationoftheflow definedbythenonnals 51
LB.6 Transversalityholds 53
LB.7 Asketchofthedefonnationargumentof[2] 55
LB.8 Someprecisionaboutthesmallnonnalsflow of[2] 61
LB.9 Appendix: TheexitsetfrominfinityintoCp 65
II IntermediateSection: Recalling the Results Described in
the Introduction, Outlining the Contentofthe Next
Sectionsand HowThese Results are Derived..••••.....71
III TechnicalStudy ofthe Critical Pointsat Infinity:
VariationalTheory without the Fredholm Hypothesis •• 75
UI.A TeueCriticalPointsatInfinity•.....••••.••••.•.....•••••••••77
x
III.A.a Onthenumberofzerosofthe v-componentof along
Wu(Voo) 77
IILA.b ThePoincare-returnmapofatruecriticalpointatinfinityXoo78
III.A.c Themodificationofthenumberofzeroson Wu(Voo):
Transmutations " 81
III.B FalseCriticalPointsatInfinityoftheSecondKind•.•.••••.•103
III.B.l Ahiddencompaniontodegeneratingperiodic
orbits 103
L~=l r2k
III.B.2 Criticalpointsof on 107
ct Qj
III.B.3 Nonnalsin andlocalparametrizationnearafalsecritical
pointatinfinityofthesecondkind 109
ct
IILB.3.a Thenonnals in whichdonotincreasethenumberof
zerosofb,andtherelatedconditions 109
Contents vii
m.B.3.b Local parametrizationnearafalse infinityofthesecond
kind 112
III.B.4 The numberofzerosontheunstablemanifolds
ofafalse criticalpointatinfinityofthesecondkind.
Theself-adjustingdirectionsofthenormalindex 119
III.B.5 Changeofpartofthe normalindexintothetangentialindex,
orvice versa, atafalse criticalpointatinfinityofthesecond
kind 123
m.B.6 Conesassociatedtoafalsecriticalpointatinfinity
ofthesecondkind 126
m.B.7 Singularities,cancellations,transmutationsalong
differentiable homotopies 127
III.B.8 More 130
III.B.8.a Moreonsingularities 130
m.B.8.b Moreontransmutations 133
m.B.9 ThePoincare-returnmapofafalsecriticalpointatinfinityof
mixedtypepreservesarea 134
III.B.I0 Topologicalremarks 136
IV Removalof(AS) 143
IV.l TheDifferenceofTopologyDuetoa FalseCriticalPointat
InfinityoftheThirdKind.••..•••.•..••..••..••••.••••..•••145
IV.2 CompletionoftheRemovalof(AS)..•••••••••.•..••••..••••151
IV.3 CriticalPointsatInfinityofMixedType•••.••••.••••.•••..• 191
IV.4 (AS)andtheCriticalPointsatInfinityoftheThirdKind
orofMixedType ..•...•...•....•..••.•••.••.••••..•••••••• 193
V Conditions (A2)-(A3)-(A4)-(A6).••.•••••••..•••••.••195
V.l AnOutlineforthe Removalof(A2)••..••.••..••••.••.••••••197
V.2 Discussionof(A3) ..•.•...•..••..••.••••••••••••..••••.•••• 207
V.3 WeakeningCondition(A4)..••.••.•..••..••..•..•.••..••••.213
V.4 RemovingCondition(A6).••.•••.•••••..••..••.••••.••••••.215
References 217
Flow Lines and Algebraic Invariants
in Contact Form Geometry
Introduction, Statement
of Results, and Discussion
of Related Hypotheses
Thismonographisrelatedtotwopreviousones,[1]and[2],inthesamedirection
and is an attempt to createa new tool for the studyofoneorseveral aspects of
thedynamicsofacontactstructureandacontactvectorfieldinthefamily which
itdefines.
In a very early work, never published but related in full detail in the first
sectionsof[1],wehad,incollaborationwithD.Bennequin,givenacontactform
a on athree-dimensionalcompactmanifold M andanonsingular vectorfield v
fJ
=
in its kernel, setup avariational problem J(J(x) ax(.i)dt) on aspaceof
Legendriancurvesx E CfJ' where
CfJ ={x E HI(Sl, M) suchthatda(x, v) ==0, a(x) =Cste > OJ.
=
Thevariationalproblem Jon CfJ' underthe hypothesis thatfi da(v, .) wasa
contactform, hadtheperiodicorbitsofthecontactvectorfieldofa,~,ascritical
points. However, J onCfJ wasill-posedinthatitsgradientwasnotFredholmand
thedecreasingflow-lines ofanyassociatedpseudo-gradientwere notcompact.
Afternoticingthesefacts, weundertookthegoaloftryingtoovercomethese
difficultiesandtounderstandthenon-Fredholmcharacterandthenoncompactness
ofthis variationalproblem.
In [1],webuilt,undersomeratherdemandinghypotheses,apseudo-gradient
Z for which we wereable to describe the noncompactness. We showed that, to
eachnoncompactflow-line ofZ, wecouldassociatealimitcurvewhichdidnot
belongto CfJ buttosomeboundaryofCfJ. Suchcurves were madeofpiecesof
~-orbitsalternatedwithpiecesof±v-orbitsrunningbetween"conjugatepoints"
(seePartIfor alldefinitions).
In [2], weimprovedonthis result: webuiltanew pseudo-gradient Zo under
thesolehypotheses:
(At)The bundledefinedbykera over Mistrivializable.
(A2) fi = da(v, .) isacontactformdefining thesameorientationasa.
Zo had an interesting property-(P) in this introduction-which could be
thoughtofasbothanalyticalandgeometrical. Namely,denotingx=a~+bvthe
A. Bahri, Flow Lines and Algebraic Invariants in Contact Form Geometry
© Birkhäuser Boston 2003
4 Introduction,StatementofResults
tangentvectortoacurvex ofCfJ, thenumberofzerosofbneverincreasedalong
thedecreasingflow-lines ofZoo In [2] asin [1],the noncompactanddecreasing
flow-linesofZocouldbetracedbacktoafamilyofcurves,ofthesametypeasthe
onesfoundforZ,onlywithmorediversity. ThisfamilyisdescribedinChapterI.A
ofthe present work. Astudy ofthese limiting curves was started in [2] and is
m
pushedmuchfurtherinPart ofthepresentwork. Atthetopologicallevel,we
indicateinsectionI.A.4cananalogybetweenthespaceofformalbarycentersover
amanifoldandthespaceoftheselimitingcurves. Thisanalogymightturnoutto
bemeaningfulin more thanonedirection, sincethespaceofformal barycenters
appearsinanatural way when the noncompactnessinYamabe-typeproblemsis
studied(see [3],forexample).
Ontheotherhand,asweindicateinI.A.4b, thespaceoftheselimitingcurves
bears an analogy with discrete models for the free loop space of a manifold
developedbytopologistsinthe 1980s(see [4], [5], [6],forexample).
1. Topological results
Themaingoalofthisbookis todraw all thepossibleconsequences,atthetopo
logicallevel, from (P).Thisideais simple: First, since Zo satisfies (P), wecan
alsothinkofJ asdefinedon
C$ = {x E CfJ suchthatthe v-componentofX, b,hasatmost2p zeros}.
WeproveinI.BthatgeneralpositionargumentsapplyforZo,Le.,thatwecan
assume that the stable and unstable manifolds ofthe critical points (at infinity)
of Zo intersect transversally while (P) holds. Therefore, we may considerthe
intersectionoperatoraofthevariationalproblemJ onC$. Thisprovidesrefined
a
versionsofthefull intersectionoperator on CfJ.
Next, weassumeanadditional hypothesis:
Let,foreachk E N*, r2k bethespaceofcurvesmadeofkpiecesof~-orbits
alternated withk piecesof±v-orbits. Weassumethefollowing:
(A3) J(x), extendedandrestrictedtoUk rZko has,oneach r2k, adecreasing
pseudo-gradientsatisfyingthePalais-Smalecondition.
(A3) willbediscussed later. Under(A3), wecanmodify Zo and incorporate
in it the pseudo-gradient defined on Uk r2k. (P) holds for this new Zoo The
noncompactdecreasing flow-lines of Zo change and become acountablesetof
isolatedasymptotesconvergingtocriticalpointsorcriticalpointsatinfinity(x~)
forwhichaMorseindexandastableandan unstablemanifoldatinfinitycanbe
defined. The definitions ofthese stable and unstable manifolds involve some
complication, due to the non-Fredholm character of this variational problem.
Thiscomplicationisdiscussedlater.
We quote now a result of [7] which states that, under the general position
assumption, if x~, a critical point (at infinity) of Morse index m, dominates
x:;'_l'acriticalpoint(atinfinity)ofMorseindexm - 1,then
Wu(x:;'_l) C Wu(x;;»
