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1、ANOBATA-TYPETHEOREMINCRGEOMETRYSONG-YINGLIANDXIAODONGWANG1.IntroductionInRiemanniangeometry,estimatesonthefirstpositiveeigenvalueoftheLaplaceoperatorhaveplayedimportantrolesandtherehavebeenmanybeautifulresults.WereferthereadertothebooksChavel[C]andSchoen-Yau[SY].Thefollowingthe
2、oremisaclassicresult.Theorem1.(Lichnerowicz-Obata)Let(Mn,g)beaclosedRiemannianmanifoldwithRic≥(n−1)κ,whereκisapositiveconstant.Thenthefirstpositiveeigen-valueofLaplaciansatisfies(1.1)λ1≥nκ.Moreover,equalityholdsiffMisisometrictoaroundsphere.Theestimateλ1≥nκwasprovedbyLichnerowicz
3、[L]in1958.Thecharacteriza-tionoftheequalitycasewasestablishedbyObata[O]in1962.Infact,hededuceditfromthefollowingmoregeneralTheorem2.(Obata[O])Suppose(Nn,g)isacompleteRiemannianmanifoldanduasmooth,nonzerofunctiononNsatisfyingD2u=−c2ug,thenNisisometrictoasphereSn(c)ofradius1/cin
4、theEuclideanspaceRn+1.InCRgeometry,wehavethemostbasicexampleofasecondorderdifferentialoperatorwhichissubelliptic,namelythesublaplacian∆b.Onaclosedpseudo-hermitianmanifold,thesublaplacian∆bstilldefinesaselfadjointoperatorwithadiscretespectrum(1.2)λ0=0<λ1≤λ2≤···withlimk→∞λk=+∞.One
5、wouldnaturallyhopethatthestudyoftheseeigenval-arXiv:1207.4033v2[math.DG]13Aug2012uesinCRgeometrywillbeasfruitfulasinRiemanniangeometry.AnanalogueoftheLichnerowiczestimateforthesublaplacianonastrictlypseudoconvexpseudo-Hermitianmanifold(M2m+1,θ)wasprovedbyGreenleafin[G]form≥3an
6、dbyLiandLuk[LL]form=2.LateritwaspointedoutthattherewasanerrorintheproofoftheBochnerformulain[G].Duetothiserror,theBochnerformulaaswellastheCR-Lichnerowicztheoremin[G,LL]arenotcorrectlyformulated.ThecorrectedstatementisTheorem3.LetMbeacompact,strictlypseudoconvexpseudo-hermitia
7、nmanifoldofdimension2m+1≥5.SupposethattheWebsterpseudoRiccicurvatureandthepseudotorsionsatisfym+12(1.3)Ric(X,X)−Tor(X,X)≥κ
8、X
9、2ThesecondauthorwaspartiallysupportedbyNSFgrantDMS-0905904.12SONG-YINGLIANDXIAODONGWANGforallX∈T1,0(M),whereκisapositiveconstant.Thenthefirstpositiveeige
10、n-valueof−∆bsatisfiesm(1.4)λ1≥κ.m+1Theestimateissharpasonecanv
