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1、LaplacianonRiemannianmanifoldsBrunoColbois1erjuin2010Preamble:Thisareinformalnotesofaseriesof4talksIgaveinCarthage,asintroductiontotheDidoConference,May24-May29,2010.Thegoalistopresentdi erentaspectsoftheclassicalquestion"HowtounderstandthespectrumoftheLaplac
2、ianonaRiemannianmanifoldthankstothegeometryofthemanifold?"The rstlecturepresentssomegeneralitiesandsomegeneralresults,thesecondlectureconcernsthehyperbolicmanifolds,thethirdlecturegivesestimatesontheconformalclass,andthelastpresentsomeestimatesforsubmanifolds
3、.Thelectureendswithsomeopenquestions.1Introduction,basicresultsandexamplesLet(M;g)beasmooth,connectedandC1Riemannianmanifoldwithboundary@M.TheboundaryisaRiemannianmanifoldwithinducedmetricgj@M.Wesuppose@Mtobesmooth.WerefertothebookofSakai[Sa]forageneralintrod
4、uctiontoRiemannianGeometryandtoBerard[Be]andChavel[Ch1]foranintroductiontospectraltheory.Forafunctionf2C2(M),wede netheLaplaceoperatororLaplacianbyf=df= divgradfwheredistheexteriorderivativeandtheadjointofdwithrespecttotheusualL2-innerproductZ(f;h)=fhdVMw
5、heredVdenotesthevolumeformon(M;g).Inlocalcoordinatesfxig,theLaplacianreads1X@p@pijf= (gdet(g)f):det(g)@xj@xii;j1Inparticular,intheEuclideancase,werecovertheusualexpressionX@@f= f:@xj@xjjLetf2C2(M)andh2C1(M)suchthathdfhascompactsupportinM.ThenwehaveGreen'sFo
6、rmulaZZdf(f;h)=dV hdAM@Mdndfwheredenotesthederivativeoffinthedirectionoftheoutwardunitnormalvectordn eldnon@ManddAthevolumeformon@M.dfInparticular,ifoneofthefollowingconditions@M=;,hj@M=0or(dn)j@M=0issatis ed,thenwehavetherelation(f;h)=(df;dh):Inthes
7、equel,wewillstudythefollowingeigenvalueproblemswhenMiscompact:{ClosedProblem:f=finM;@M=;;{DirichletProblemf=finM;fj@M=0;{NeumannProblem:dff=finM;()j@M=0:dnWehavethefollowingstandardresultaboutthespectrum,see[Be]p.53.Theorem1.LetMbeacompactmanifoldwithbo
8、undary@M(eventuallyempty),andconsideroneoftheabovementionedeigenvalueproblems.Then:1.Thesetofeigenvalueconsistsofanin nitesequence0<123:::!1,where0isnotaneigenvalueintheDirichletpro
