Hofer-Zehnder semicapacity of cotangent bundles and symplectic submanifolds

arXiv:math/0303230v3 [math.SG] 18 Aug 2003HOFER-ZEHNDERSEMICAPACITYOFCOTANGENTBUNDLES

ANDSYMPLECTICSUBMANIFOLDS

LEONARDOMACARINI

Abstract.WeintroducetheconceptofHofer-ZehnderG-semicapacity(orG-

sensitiveHofer-Zehndercapacity)andprovethatgivenageometricallybounded

symplecticmanifold(M,ω)andanopensubsetN⊂MendowedwithaHamilton-

ianfreecircleactionϕthenNhasboundedHofer-ZehnderGϕ-semicapacity,where

Gϕ⊂π1(N)isthesubgroupgeneratedbythehomotopyclassoftheorbitsofϕ.In

particular,NhasboundedHofer-Zehndercapacity.

Wegivetwotypesofapplicationsofthemainresult.Firstly,weprovethat

thecotangentbundleofacompactmanifoldendowedwithafreecircleactionhas

boundedHofer-Zehndercapacity.Inparticular,thecotangentbundleT∗Gofany

compactLiegroupGhasboundedHofer-Zehndercapacity.Secondly,weconsider

Hamiltoniancircleactionsgivenbysymplecticsubmanifolds.Forinstance,weprove

thefollowinggeneralizationofarecentresultofGinzburg-G¨urel[7]:almostalllevels

ofafunctiondefinedonaneighborhoodofaclosedsymplecticsubmanifoldSina

geometricallyboundedsymplecticmanifoldcarrycontractibleperiodicorbitsofthe

Hamiltonianflow,providedthatthefunctionisconstantonS.

1.Introduction

Wewillconsiderheretheproblemoftheexistenceofperiodicorbitsonprescribed

energylevelsofHamiltoniansystems.Anapproachtothisproblemwastheintroduc-

tionofcertainsymplecticinvariantsrelatedtosymplecticrigidityphenomena.More

precisely,H.HoferandE.Zehnderintroducedin[9]thefollowingdefinition:

Definition1.1.Givenasymplecticmanifold(M,ω),definetheHofer-Zehnderca-

pacityofMby

c

HZ(M,ω)=sup

H∈Ha(M,ω)maxH,

whereH

a(M,ω)isthesetofadmissibleHamiltoniansHdefinedonM,thatis,

•H∈H(M)⊂C∞(M,R),whereH(M)isthesetofpre-admissibleHamilto-

niansdefinedonM,thatis,Hsatisfiesthefollowingproperties:0≤H≤

󰀟H󰀟:=maxH−minH,thereexistanopensetV⊂MsuchthatH|

V≡0

andacompactsetK⊂M\∂MsatisfyingH|

M\K≡󰀟H󰀟;

•everynon-constantperiodicorbitofX

Hhasperiodgreaterthan1.2LEONARDOMACARINI

WewillalsoconsiderhereamodifiedHofer-ZehndercapacityintroducedbyD.

McDuffandJ.Slimowitz[16]sensitivetothepresenceofovertwistedcriticalpoints.

Itisgivenby

󰀅c

HZ(M,ω)=sup

H∈󰀅Ha(M,ω)maxH,

whereH∈󰀅H

a(M,ω)ifHisadmissibleandhasnoovertwistedcriticalpoints,that

is,

•thelinearizedflowatthecriticalpointsofHhasnonon-trivialperiodicorbits

ofperiodlessthan1.

Remark.Itisanopenquestionwhetherthecapacitiesc

HZand󰀅c

HZcoincide.

Itiseasytoprovethatifasymplecticmanifold(M,ω)hasboundedHofer-Zehnder

capacity,thatis,ifc

HZ(U,ω)<∞foreveryopensubsetU⊂Mwithcompact

closure,thengivenanyHamiltonianH:M→Rwithcompactenergylevels,there

existsadensesubsetΣ⊂H(M)suchthatforeverye∈Σtheenergyhypersurface

H−1(e)hasaperiodicsolution.Actually,ifthereexistsaneighborhoodofanenergy

hypersurfaceH−1(e

0)withfiniteHofer-Zehndercapacitythenthereexistsǫ>0anda

densesubsetG⊂(e

0−ǫ,e

0+ǫ)suchthatH−1(e)hasperiodicorbitsforeverye∈G.

Thisresultwasimprovedin[14],whereitisshowedthatthereareperiodicorbitson

H−1(e)foralmostalle∈(e

0−ǫ,e

0+ǫ)withrespecttotheLebesguemeasure.The

sameholdsfor󰀅c

HZ(M,ω).

LetusconsiderherearefinementoftheoriginalHofer-Zehndercapacitybycon-

sideringperiodicorbitswhosehomotopyclassiscontainedinagivensubgroupGof

π

1(M).Moreprecisely,wedefinetheHofer-Zehndersemicapacityasfollows:

Definition1.2.Givenasymplecticmanifold(M,ω)andasubgroupG⊂π

1(M)

definetheHofer-ZehnderG-semicapacity(orG-sensitiveHofer-Zehndercapacity)of

Mby

cG

HZ(M,ω)=sup

H∈HGa(M,ω)maxH,

whereHG

a(M,ω)isthesetofG-admissibleHamiltoniansdefinedonM,thatis,

•H∈H(M),thatis,Hispre-admissible(seethedefinition1.1);

•everynonconstantperiodicorbitofX

HwhosehomotopyclassbelongstoG

hasperiodgreaterthan1.

Wedefine󰀅cG

HZ(M,ω)analogouslyconsideringG-admissibleHamiltonianswithout

overtwistedcriticalpoints.

Remark.Wecallitasemicapacitybecausegivenasymplecticembeddingψ:

(N,τ)→(M,ω)suchthatdimN=dimMitcannotbeexpected,ingeneral,that

cG

HZ(N,τ)≤cψ∗G

HZ(M,ω),whereψ

∗:π

1(N)→π

1(M)istheinducedhomomorphism

onthefundamentalgroup.However,wecanstatethefollowingweakmonotonicity

property(foraproofsee[13]):givenasymplecticembeddingψ:(N,τ)→(M,ω)

suchthatdimN=dimMthen

cψ−1

∗H

HZ(N,τ)≤cH

HZ(M,ω),HOFER-ZEHNDERSEMICAPACITYOFCOTANGENTBUNDLES3

foreverysubgroupH⊂π

1(M).Inparticular,ifψ

∗isinjectivewehavethat

cG

HZ(N,τ)≤cψ∗G

HZ(M,ω).

Notethatobviously,

c

HZ(M,ω)=cπ1(M)

HZ(M,ω)≤cG

HZ(M,ω),

foreverysubgroupG⊂π

1(M).Moreover,itcanbeshowthat,liketheHofer-

Zehndercapacity,iftheHofer-ZehnderG-semicapacityisboundedthenthereare

periodicorbitswithhomotopyclassinGonalmostallenergylevelsforeveryproper

Hamiltonian[14].Definition1.3.Asymplecticmanifold(M,ω)iscalledgeometricallyboundedifthere

existsanalmostcomplexstructureJonMandaRiemannianmetricgsuchthat

•Jisuniformlyω-tame,thatis,thereexistpositiveconstantsc

1andc

2such

that

ω(v,Jv)≥c

1󰀟v󰀟2and|ω(v,w)|≤c

2󰀟v󰀟󰀟w󰀟

foralltangentvectorsvandwtoM;

•thesectionalcurvatureofgisboundedfromaboveandtheinjectivityradius

isboundedawayfromzero.

Closedsymplecticmanifoldsareclearlygeometricallybounded;aproductoftwo

geometricallyboundedsymplecticmanifoldsisalsosuchamanifold.Itwasshowed

in[5,11]thatthecotangentbundleofacompactmanifoldendowedwithanytwisted

symplecticformisgeometricallybounded.Atwistedsymplecticstructureonacotan-

gentbundleT∗Misasymplecticformgivenbyω

0+π∗Ω,whereω

0isthecanonical

symplecticform,π:T∗M→MisthecanonicalprojectionandΩisaclosed2-form

onM.

Definition1.4.Givenasymplecticmanifold(M,ω)defineitsindexofrationality

by

m(M,ω)=inf󰀆󰀇

[u]ω;[u]∈π

2(M)satisfies󰀇

[u]ω>0󰀈

.

Iftheabovesetisempty,wedefinem(M,ω)=∞.Ifm(M,ω)>0wecall(M,ω)a

rationalsymplecticmanifold.

ThemainresultherestatesthattheexistenceofafreeHamiltoniancircleaction

impliestheboundednessoftheHofer-Zehndersemicapacity.Itremovesatechnical

hypothesisinaprevioustheoremin[13].

MainTheorem.Let(M,ω)beageometricallyboundedsymplecticmanifoldand

N⊂ManopensubsetthatadmitsafreeHamiltoniancircleactionϕ.Then,given

anyopensubsetUi֒→Nwithcompactclosure,

󰀅ci−1

∗Gϕ

HZ(U,ω)<∞,

wherei

∗:π

1(U)→π

1(N)istheinducedhomomorphismonthefundamentalgroup

andG

ϕ⊂π

1(N)isthesubgroupgeneratedbytheorbitsofthecircleaction.Moreover,

ifMisrational,thenci−1

∗Gϕ

HZ(U,ω)<∞.