EXTRINSIC UPPER BOUNDS FOR THE FIRST EIGENVALUE OF ELLIPTIC OPERATORS

Jean - Franc¸Ois Grosjean

Description

Niveau: Supérieur, Doctorat, Bac+8
EXTRINSIC UPPER BOUNDS FOR THE FIRST EIGENVALUE OF ELLIPTIC OPERATORS Jean-Franc¸ois GROSJEAN 2002 Address: GROSJEAN Jean-Francois, Institut Elie Cartan (Mathematiques), Universite Henri Poincare Nancy I, B.P. 239, F-54506 VANDOEUVRE-LES-NANCY CEDEX, FRANCE. E-mail: Abstract: We consider operators defined on a Riemannian manifold Mm by LT (u) = ?div(T?u) where T is a positive definite (1, 1)-tensor such that div(T ) = 0. We give an upper bound for the first nonzero eigenvalue ?1,T of LT in terms of the second fundamental form of an immersion ? of Mm into a Riemannian manifold of bounded sectional curvature. We apply these results to a particular family of operators defined on hypersurfaces of space forms and we prove a stability result. Key words: r-th mean curvature, Reilly's inequality MSC 1997: 53 A 10, 53 C 42 1

• divergence tensor


• sectional curvature


• let


• optimal upper bound


• manifold isome- trically


• tensor such


• estimates still


• dimensional riemannian


• manifold


• finite positive

Sujets

• Grosjean
• Poincaré
• Sectional curvature
• Manifold

Informations

• Publié par: profil-zyak-2012
• Nombre de lectures: 8
• Langue: English

Extrait

EX E T IG RI E N N S V IC AL U U P E PE O R F B EL O L U I N PT D I S C F O O P R E T R H A E TO FI R R S ST Jean-Franc¸oisGROSJEAN 2002 N A a d n d c r y es I s , : B G .P R . O 2 S 3 J 9, E F A - N 54 J 5 e 0 a 6 n- V F A ra N n D co O is E , U In V st R it E u - t LE E ´ S li - e N C A a N rt C a Y n( C M E a D th E e´ X m , a F ti R q A ue N s) C , E U . niversite´HenriPoincare´ E-mail:[e-mail] Abstract:WeconsideroperatorsdefinedonaRiemannianmanifoldM m byL T (u)= − div(T r u)where Tisapositivedefinite(1,1)-tensorsuchthatdiv(T)=0.Wegiveanupperboundforthefirstnonzero eigenvalueλ 1,T ofL T intermsofthesecondfundamentalformofanimmersionφofM m intoaRiemannian manifoldofboundedsectionalcurvature.Weapplytheseresultstoaparticularfamilyofoperatorsdefined onhypersurfacesofspaceformsandweproveastabilityresult. Keywords :r-thmeancurvature,Reilly’sinequality MSC1997:53A10,53C42 1

1Introduction Let(M m ,g)beacompact,connectedm-dimensionalRiemannianmanifold.Inthis paper,weareinterestedinextrinsicupperboundsforthefirstnonzeroeigenvalueof ellipticoperatorsdefinedon(M m ,g)(i.e.intermsofthesecondfundamentalformof anisometricimmersionof(M m ,g)intoann-dimensionalRiemannianmanifold(N n ,h)). TheellipticsecondorderdifferentialoperatorsL T ,whichweareinterestedin,areofthe mrof L T u= − div M (T r M u) whereu ∈ C ∞ (M),Tisa(1,1)-tensoronM(whichwillbedivergence-freeandsymmet- ric),anddiv M and r M denoterespectivelythedivergenceandthegradientofthemetric g.Inthesequel,wewilldenotebyλ 1,T ,thefirstnonzeroeigenvalueofsuchoperatorL T . WhenTistheidentity,L T =L Id isnothingbuttheLaplaceoperatorof(M m ,g).In thiscase,itiswellknownthatif(M m ,g)isisometricallyimmersedinasimplyconnected spaceformN n (c)(c=0,1, − 1respectivelyfortheEuclideanspaceIR n ,thesphereSI n or thehyperbolicspaceIH n ),thenwehavethefollowingestimateofλ 1 =λ 1,Id intermsof thesquareofthelengthofthemeancurvature Z
� λ 1 V(M) ≤ m | H | 2 +cdv g (1) M wheredv g andV(M)denoterespectivelytheRiemannianvolumeelementandthevolume of(M m ,g)andwhereHdenotesthemeancurvatureoftheimmersionof(M m ,g)into N n (c).Furthermore,theequalityin(1)occursifandonlyif(M m ,g)isimmersedasa minimalsubmanifoldofsomegeodesichypersphereofN n (c).Forc=0,thisinequality wasprovedbyReilly([17])andcaneasilybeextendedtothesphericalcasec=1by consideringthecanonicalembeddingofSI n inIR n+1 andbyapplyingtheinequality(1)for c=0totheobtainedimmersionof(M m ,g)inIR n+1 .Forimmersionsof(M m ,g)inthe hyperbolicspaceIH n ,Heintze([14])firstprovedanL ∞ equivalentof(1)andconjectured (1)whichwasfinallyobtainedbyElSoufiandIlias([9]).Notethat,theestimatesshown in[14]and[9]aregivenforimmersionsof(M m ,g)inaspacewhichisnotnecessarlyof constantsectionalcurvature. Later,theseestimateswereextendedtomoregeneraloperatorscalledL r (0 ≤ r ≤ n) definedonhypersurfaces(M m ,g)ofN m+1 (c).Letusfirstdefinetheseoperators.Let φbeanisometricimmersionof(M m ,g)intoN m+1 (c)anddenotebyAitsshape(or Weingarten)operator.Foranyintegerr ∈{ 0,...,n } ,the(1,1)-tensorsT r ofNewtonare definedinductivelyby:T 0 =IdandT r =S r Id − AT r − 1 ,whereS r isther-thelementary symmetricfunctionoftheeigenvaluesofA(i.e.theprincipalcurvatures).NotethatT r isafreedivergencetensorbecausetheambientspace � is � of
c
onstantcurvature(seefor instance[19]).Ther-thmeancurvatureofφisH r =1/ r m S r .Now,theoperatorL r 2