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ONTHE SPECTRUMOF X¯-BOUNDED MINIMALSUBMANIFOLDS 9 0 Isabel M.C. Salavessa 0 2 n a J CentrodeF´ısicadasInteracço˜esFundamentais,InstitutoSuperiorTećnico,Tech- 4 nicalUniversityofLisbon,Edif´ıcioCieˆncia,Piso3,Av.RoviscoPais,P-1049-001 1 Lisboa,Portugal; e-mail:[email protected] ] G D . h Abstract t a We prove, under a certain boundedness condition at infinity of a (X¯ ,X¯ ) m ⊤ ⊥ component of the second fundamental form, the vanishing of the essential [ spectrum ofacomplete minimalX¯-bounded andX¯-properly immersedsub- 3 manifold on a Riemannian manifold endowed with a strongly convex vec- v 6 tor field X¯. The same conclusion also holds for any complete minimal h- 4 bounded and h-properly immersed submanifold that lies in a open set of a 2 Riemannian manifold M supporting a nonnegative strictly convex function 1 . h. Thisextendsarecent resultofBessa,Jorge andMontenegro onthespec- 1 0 trum of Martin-Morales minimal surfaces. Our proof uses as main tool an 9 extensionofBarta’stheorem givenin[2]. 0 : v i 1 Introduction and main results X r a SinceCalabi in 1965[4] conjectured that completeminimalhypersurfaces in Eu- clidean spaces are unbounded, some answers have been given, with a positive answer by Colding and Minicozzi [6] for the case of embedded surfaces, and a negative answer with the counterexamples given by Nadirashvili [14] and by Martin and Morales [12, 13] for the case of immersed nonembedded surfaces. MSC0 2000:Primary:53C40;58C40 KeyWords: spectrum,minimalsubmanifold,convexfunction PartiallysupportedbyFCTthroughthePlurianualofCFIF. 1 This conjecture also motivates many other related problems in more general ambient spaces, for instance, on the topological and geometrical properties of minimal submanifolds that are bounded or not, or on the search of conditions for a submanifoldto be unbounded. In [3] the structure of the spectrum of the Martin- Morales surfaces is studied, namely it is proved that complete bounded minimal properlyimmersedsubmanifoldsoftheunitopenballofRn musthavepurepoint spectrum. In this note we extend the above result of Bessa, Jorge and Montenegro to an ambient space carrying an almost conformal vector field X¯, a concept introduced in ([16, 17]). On a Riemannian (m+n)-dimensional manifold (M¯,g¯) we say a vectorfield X¯ is almostconformal if 2a g¯ L g¯ 2b g¯ (1) X¯ ≤ ≤ where +¥ b a >0 are constants, and L g¯(Y¯,Z¯)=g¯((cid:209)¯ X¯,Z¯)+g¯((cid:209)¯ X¯,Y¯), X¯ Y¯ Z¯ where(cid:209)¯ is≥theL≥evi-Civitaconnectionof(M,g¯). Ifweallowb =+¥ ,inthiscase X¯ isnamed bystronglyconvex. An exampleof almostconformalvectorfield in a completeRiemannian manifold M is the position vector field 1(cid:209)¯r2 =r ¶ on a geodesic ball of M of radius 2 ¶ r R and center p¯ that does not intercept the cut locus at p¯ and √k +R <p /2 with k + = k +(R) = max 0,sup K¯ , where K¯ are the sectional curvatures of M¯ { BR(p¯) } and r is the distance function on M to a given point. In this case a and b are well defined functions a =a k +(R), b =a k (R), of R, k +, and k − =k −(R)= − min 0,inf K¯ where { BR(p¯) } R√k cot(√k R) for0 R<p /2√k , when k >0 a k (R)= 1 for0≤R<+¥ , when k =0 (2)  R√ k coth(√ k R) for0≤R<+¥ , when k <0. − − ≤  AstrictlyconvexfunctionfonM withHessf a g¯ definesastronglyconvexvec- tor field (cid:209)¯ f. Positive homotheticnon-Killing≥vector fields are almost conformal. In Rm+n the position vector field X¯ =x is such an example. A particular feature x of strongly convex vector fields, is that the norm X¯ must take its maximum on k k the boundary of compact domains (see proposition 1). Therefore X¯ cannot be globallydefined on acompact manifoldM withoutboundary. Strongly convex vector fields have a role on isoperimetric inequalities for an immersedm-dimensionalsubmanifoldF :M M,m 2,involvingthethemean curvatureH. TheCheegerconstantofM isde→finedby≥h(M)=inf A(¶ D)/V(D), D 2 where D runs over all compact domains D of M with picewise smooth boundary ¶ D M of respective volume V(D) and area A(¶ D). We recall the following ⊂ inequality[11]: 1 1 (sup X¯ ) 1 h(M)+sup H (3) k Fk − ≤ a (cid:18)m k k(cid:19) M M whereX¯ denotesX¯ alongF. LetX¯ andX¯ denotetheorthogonalprojectionsof F ⊤ ⊥ X¯ onto TM, and the normal bundleNM respectively. We remark that, following F theproofin [11]wesee thatifF isminimalwehaveasharperinequality: 1 1 (sup X¯ ) 1 h(M). (4) k ⊤k − ≤ a m M We notethat X¯ (X¯ resp.) cannot vanishidentically for any (minimalresp.) im- F ⊤ mersion F (see lemmas 1 and 2). In the case M is the Euclidean space with the position vector field, X¯ X¯ = F . This leads to the following conclu- ⊤ F k k ≤ k k k k sion: Theorem 1 ([11]). If X¯ is a strongly convex vector field on a neighbourhood of a minimal submanifold F : M M¯ with zero Cheeger constant, then X¯ is ⊤ unbounded. In theparticularcase→M =Rn+m, F isunbounded. Werecall thefollowinginequalitydueto Cheeger[5], h2(D) 4l (D) ≤ where l (D) is the fundamental tone of a normal domain D in M. For normal boundeddomains,l (D)isthefirsteigenvaluefortheboundaryDirichletproblem. The Rayleigh characterization of the fundamental tone of any open domain D of M isgivenby (cid:209) f 2 l (D)=inf Dk k : f L2 (D) (cid:26)R f2 ∈ 1,0 (cid:27) D whereL2 (D)isthecompletationofCR ¥ (D)forthenorm f 2= f 2+ (cid:209) f 2. 1,0 0 k k M k k Thus, if M is complete noncompact, l (M) = lim l (D ) and hR(M) h(D ), R R R ≤ where D is an exhaustion sequence of bounded domains of M with smooth R boundary in M. Therefore, from the above inequalities we have the following 3 estimate for M a bounded domain (possibly with boundary) or a complete Rie- mannianmanifold 1 2 (sup X¯ ) 1 ( l (M)+sup H ) k Fk − ≤ a m k k M p M 1 2 (sup X¯ ) 1 l (M), ifM isminimal. (5) k ⊤k − ≤ a m M p Definition 1. Given a vector field X¯ of M, an immersed submanifoldF :M M is said X¯-bounded if sup X¯ <+¥ . If sup X¯ is not achieved, then→F is M ⊤ M ⊤ k k k k saidX¯-proper,if X¯ :M [0,sup X¯ )is apropermap. ⊤ M ⊤ k k → k k We will see in proposition 1 that if M is minimal and X¯ is strongly convex, then sup X¯ is not achieved (in M) if condition (6) below holds. Note that if X¯ M ⊤ k k is the position vector field of M, X¯ X¯ = r(F(p)). This implies X¯- p⊤ F(p) k k ≤ k k boundedness is a weaker concept then the usual boundedness of M in M. For example, the spiral curve in R2, g (t) = aetb(cos(eabt),sin(eabt)) with a > 1 and b > 0 constants, is X¯-bounded but unbounded in the usual sense. On the other hand X¯-properness might be a stronger concept than the usual properness of an immersion. Wealso remark thatifX¯ =0along allM, then (cid:209)¯r restricted toM is ⊥ a vector field on M. If r is the distance function on M from a fixed point p M, weseethat(unit)geodesics ofM startingat p (thatare theintegralcurves o∈f(cid:209)¯r) liein M. In thiscase n=0. Nextwestateourmain theorems: Theorem 2. LetF :M M¯ beacompleteminimalimmersionthatisX¯-bounded → with sup X¯ =R, where X¯ is a stronglyconvex vector field of M defined on a M ⊤ k k neighbourhoodofM, then: (1) 2 l (M) h(M) ma . ≥ ≥ R p (2)Furthermore,ifthesecond fundamentalformBof M satisfiesatpoints p M ∈ with X¯ sufficientlyclosetoR, ⊤ k k g¯(B(X¯ ,X¯ ),X¯ ) a X¯ 2, (6) ⊤ ⊤ ⊥ ′ ⊤ | |≤ k k forsomenonnegativeconstanta <a ,andifM isX¯-proper,thenthespectrumof ′ M isa purepointspectrum. 4 The condition (6) does not mean B is bounded, even in the case X¯ is the po- ¶ k k sition vector field r . In theorem 5 (section 2) we will see that boundedness ¶ r of the second fundamental form is, in general, not a compatible condition with the boundedness of a complete minimal submanifold, for ambient spaces with sectional curvature bounded from below. Moreover, for the particular case of X¯ being the gradient of a nonnegativeconvex smooth function h:M [0,+¥ ) we → canremovetheboundednesscondition(6)oftheorem2,ifweadaptourdefinition of boundedness and of properness: F is h-bounded if sup h F =R<+¥ , and M ◦ is h-proper if sup h F is not achieved and h F :M [0,R) is a proper func- M ◦ ◦ → tion. We also will see in proposition1 that sup h F cannot be achieved for F a M ◦ minimalimmersion. Theorem 3. Let h : M [0,+¥ ) be a nonnegative convex smooth function and → F : M M¯ a complete minimal immersion that is h-bounded. If F is h-proper, → thenthespectrumofM is apurepointspectrum. The abovecase contains the next example, when h= 1r2, where r is the distance 2 functiontoapoint p¯ inM. NotethatifF ish-bounded,thenitisalsoX¯-bounded, for X the position vector field, and the concept of h-bounded (h-porper resp.) is equivalent to usual boundedness (properness resp.). Next corollary is a corollary oftheorem2 (1)and theorem3: Corollary 1. If F : M M is a complete bounded minimal submanifold with → F(M) lying in a open geodesic ball B (p¯) of M, and R is in the conditions given R in (2), then 2 l (M) h(M) ma , where a =a (R). Furthermore, if F is a ≥ ≥ R k+ properimmerpsionintoB (p¯), thenthespectrumof M is a purepointspectrum. R Corollary2. If F :M M isacompleteboundedminimalsubmanifoldproperly → immersed in B (p¯), and M is a complete Riemannian manifold with K¯ 0, then R ≤ 2 l (M) h(M) m andthespectrumofM is apurepointspectrum. ≥ ≥ R p The later corollaries are straightforward generalizations of [3]. Donnelly in [7] proved the existence of a non-empty essential spectrum for negatively curved manifolds under certain conditions. This result and corollary 2 gives next corollary: Corollary3. ThereisnocompletesimplyconnectedminimalsurfaceF :M2 M → properly immersed into a geodesic ball B (p¯) of a space form M of constant R sectionalcurvatureK¯ <0,andsatisfying B 2 catinfinity,foranynonnegative k k → finiteconstantc. 5 As we have announced above, in theorem 5 we will see this conclusion can be extended to a considerably more general setting, where we do not need to use spectral theory to proveit,but a generalized Liouville-typeresult dueto Ranjbar- Motlagh[15]. An application of a hessian comparison theorem for the distance function to a totallyconvexsubmanifoldduetoKasue[10]giveus thefollowingtheorem: Theorem 4. Let M be a connected complete Riemannian manifold with nonnegative sectional curvature and S a totally convex submanifold of dimension d n that is a closed subset of M, and let h = 1r 2, where r is the distance funct≥ion 2 in M to S . If F :M M is a complete minimal immersed submanifold such that for any p M F 1→(S ), (s (l)) 2 a , where 0<a 1 is a constant and ∈ \ − k F′(p) ⊤k ≥ ≤ s : [0,l] M is the unique geodesic normal to S that satisfies s (0) S F(p) F(p) → ∈ ands (l)=F(p), then: F(p) (1) 2 l (M) h(M) ma . In particular, if M has zero Cheeger constant, thenrpF isun≥bounded≥. supMr ◦F ◦ (2) If M is h-bounded and h-properlyimmersed, then M has pure point spectrum only. In the last section we apply this general result to submanifolds of a product of Riemannianmanifolds. 2 Some inequalities for minimal submanifolds LetX¯ bean almostconformal vectorfield ofM, and F :M M an immersionof a m-dimensionalsubmanifold with second fundamental for→m B: 2TM NM, → where NM is the normal bundle of M. We give to M the inducJed Riemannian metric g = F g¯ and the corresponding Levi Civita connection (cid:209) . We denote by ∗ ( ) and ( ) the orthogonal projections of T M onto T M dF (T M) and ⊤ ⊥ F(p) p p p · · ≡ NM respectively. We have for X,Y vector fields on M, (cid:209) Y = ((cid:209)¯ Y) and p X X ⊤ B(X,Y)=((cid:209)¯ Y) . The mean curvature of M is the normal vector given by H = X ⊥ 1trace B. TheprojectionX¯ defines avectorfield onM, andX¯ asectionofthe m g ⊤ ⊥ normalbundle. SinceX¯ =X¯ +X¯ , an elementary computationgives F ⊤ ⊥ Lemma 1. For Y,Z T M, L g(Y,Z) =L g¯(Y,Z)+2g¯(B(Y,Z),X¯ ). In par- p X¯ X¯ ⊥ ticular,X¯ cannotva∈nisheveryw⊤here inanyopendomainof M. F 6 Lemma 2. (1) ma +mg¯(H,X¯ ) div (X¯ ) mb +mg¯(H,X¯ ). If F is mini- ⊥ g ⊤ ⊥ mal then ma div (X¯ ) mb , a≤nd X¯ cann≤ot vanish everywhere in any open g ⊤ ⊤ ≤ ≤ domainofM. (2) g((cid:209) X¯ ,X¯ ) a X¯ + 1 g¯(B(X¯ ,X¯ ),X¯ ). k ⊤k ⊤ ≥ k ⊤k kX¯⊤k ⊤ ⊤ ⊥ Proof. Let e be an o.n. basis of T M. At p, div (X¯ )=(cid:229) 1L g(e,e), and an i p g ⊤ i 2 X¯⊤ i i applicationofpreviouslemmagives(1)as well (2)since g((cid:209) X¯ ,X¯ )=(cid:229) 1 g((cid:209) X¯ ,X¯ )g(e,X¯ )= 1 L g(X¯ ,X¯ ). k ⊤k ⊤ X¯ ei ⊤ ⊤ i ⊤ 2 X¯ X¯⊤ ⊤ ⊤ i k ⊤k k ⊤k Proposition1. If X¯ isstronglyconvex, then: (1) For any bounded domain D of M the norm X¯ takes its maximum on the boundary¶ D. k k (2) IfF isaminimalimmersionand(6)holds,thenthesupremumof X¯ cannot ⊤ k k beachieved. In particularM cannotbecompact withoutboundary(closed). (3)IfX¯ =(cid:209) h forasmoothnonnegativeconvexfunctionh:M RandF :M → → M is a minimal submanifold, then the supremum of h F cannot be achieved. In ◦ particularM cannotbeclosed. Proof. From the inequality g¯((cid:209)¯ X¯ 2,X¯) = 2g¯((cid:209)¯ X¯,X¯) 2a X¯ 2, all critical X¯ k k ≥ k k points of X¯ 2 are vanishing points. This proves (1). To prove (2) we assume a k k maximumpoint p of X¯ exists. Then at p we may take e =X¯ / X¯ , and 0 ⊤ 0 1 ⊤ ⊤ k k k k wehaveby lemma1 and(6) 0 = (cid:209) X¯ 2 2 =4(cid:229) g((cid:209)¯ X¯ ,X¯ ) 2 4 g((cid:209)¯ X¯ ,X¯ ) 2 X¯ 2 k k ⊤k k i| ei ⊤ ⊤ | ≥ | X¯⊤ ⊤ ⊤ | k ⊤k− = (L g(X¯ ,X¯ ))2 X¯ 2 (2a X¯ 2+2g¯(B(X¯ ,X¯ ),X¯ ))2 X¯ 2 X¯ ⊤ ⊤ ⊤ − ⊤ ⊤ ⊤ ⊥ ⊤ − ⊤ k k ≥ k k k k C2 ≥ where C = 2(a a ), what is impossible. Finally we prove (3). A maximum ′ − point p ofh F satisfiesD (h F)(p ) 0,what contradicts 0 0 ◦ ◦ ≤ D (h F) =(cid:229) (Hessh) (dF(e),dF(e))+mg¯((cid:209)¯h ,H) ma . (7) p F(p) i i F(p) ◦ ≥ i 7 Theorem 5. If K is bounded from bellow, and F : M M is any complete im- → mersed minimal submanifold with bounded second fundamental form, then for anynonnegativestrictlyconvex functionh:M [0,+¥ )defined ina neighbour- → hoodofF(M),F ish-unbounded. Proof. Let us assumethere exists a complete h-bounded immersionwith B 2 k k ≤ b,banonnegativeconstant. By Gaussequation,theRicci tensorofM isbounded frombelow. Indeed, ifY T M isaunitvector, p ∈ Ricci(Y,Y) = (cid:229) K¯(Y,e )+mg¯(H,B(Y,Y)) g¯(B(Y,e),B(Y,e)) i i i i − m(inf K¯) mb b. ≥ M − − Furthermore, (7) holds for F minimal imersion. Then theorem 2.1 of [15] gives h F(p) us limsup C, where C is a positive constant that depends on m,b,a andrMa(pl)o→w+e¥r br◦Mo(upn)d≥of K¯. This contradicts the assumption of h F to be ◦ bounded. Bessa and Montenegro defined in [2] a quantity on a domain D (bounded or not) ofM,that herewedenotebyc(D) c(D)=sup inf(div X X 2) g X (cid:18) D −k k (cid:19) whereX runsoverall vectorfieldsonD locallyintegrableandwithaweak diver- gence. Wedenotebyc(X)=div X X 2. g −k k Proposition 2 ([2]). l (D) c(D), with equality if D has compact closure with ≥ smoothboundary. Assumesup X¯ =R<+¥ and (6)holds. SetC=2(a a ). Foreach e >0 M ⊤ ′ k k − sufficientlysmallconstantweconsiderthedomain De = p M :R2 > X¯⊤ 2 >R2 e 2 . { ∈ k k − } Proposition3. IfF isaminimalsubmanifoldand(6)holds,thenforany0<e <R sufficientlysmall, mCa l (De ) . ≥ e 2 8 Proof. Wedefinethefunction f :[√R2 e 2,R) [e ,+¥ ), f(s)= C ,andthe − → R2 s2 smooth vector field on De , X = f(t)X¯⊤, where t = X¯⊤ . Using le−mma 2, we k k have c(X) = f(t)div (X¯ )+g((cid:209) (f(t)),X¯ ) f(t)2t2 g ⊤ ⊤ − 1 f(t)ma + f (t)(a t+ g(B(X¯ ,X¯ ),X¯ )) f2(t)t2. ′ ⊤ ⊤ ⊥ ≥ t − Notethat f (s)and f2(s)gofasterto+¥ then f(s),whens R. Thenwehaveto ′ require f (t)(a t+1g(B(X¯ ,X¯ ),X¯ )) f2(t)t2 0, that→holdsundercondition ′ t ⊤ ⊤ ⊥ − ≥ (6). In thiscase, Cma Cma c(X) . ≥ R2 t2 ≥ e 2 − Nowproposition2givesthelowerboundforl (De ). 3 Proof of theorems 2 and 3 Let M be a complete noncompact m-dimensional Riemannian manifold, with Laplacian operator D acting on the domain D of L2(M), where D f L2 for any f D. The spectrum of D decomposes as s (M) = s (M) s ∈(M) p ess [l (M),∈¥ ), where s (M) is the−pure point spectrum of isolated finite∪multiplici⊂ty p eigenvalues, and s (M) is the essential spectrum. The decomposition principle ess of [8] states that M and M K have the same essential spectrum, as long as K is a \ compactdomainofM withboundary. Proof of theorem 2. (1) is immediate from (4) and the Cheeger inequality. (2) We can take a sequence e 0 such that R2 e 2 are regular values of X¯ . k k F⊤ → − k k SinceF isX¯-proper,thesetsKe =M De parecompactwithsmoothboundary. As k \ k in [3] we prove the theorem by showing that l (De ) +¥ when k +¥ , what provesthat s (M)=0/. Thisisthecaseby proposkiti→on3. Proo→fof theorem ess 3. In this case we take the domain of M, De = p M :R>h F >R e , and thevectorfielddefinedonDe givenbyX =(cid:209) (h{ F∈)/(R h F◦). Then−usin}g(7), ◦ − ◦ D (h F)(p) (cid:229) i(Hessh)F(p)(dF(ei),dF(ei)) ma c(X) = ◦ = , p (R h(F(p))) (R h(F(p))) ≥ e − − and sol (De ) +¥ whene 0. → → 9 Proof of corollary 3 Assume such immersion exists with B 2 c at infinity, k k → c 0afiniteconstant. BytheGaussequationthesectionalcurvatureofM satisfy ≥ K = K¯ B 2. Then M has negative sectional curvature and K K¯ c < 0 at −k k → − infinity. By a result of Donnelly [7] the essential spectrum of M consists of the halfline[( K¯ +c)/4,+¥ ) contradictingcorollary 2. − 4 Ambient space with a totally convex set Definition2. (1)WesayavectorfieldX¯ ofM isalmosttrace-conformal(strongly trace-convexresp.) alongM if2ma Trace F L g¯ 2b m(withb =+¥ resp.), g ∗ X¯ whereb a >0 areconstants. ≤ ≤ ≥ (2) Wesaythatafunctionh:M [0,+¥ )isstrictlytrace-convexalongM iffor → somepositiveconstanta , Trace F (Hessh) ma . g ∗ ≥ It iselementaryto verifynexttheorem,followingthepreviousproofs: Theorem 6. In the weaker conditions of definitions 2 and 1, the inequality (4) stillholdsaswell theconclusionsin theorems1, 2 and3. AsubsetS ofM issaidtobetotallyconvexifitcontainsanygeodesicconnecting two points of S . If S is a submanifold that is a closed subset of M, the hessian of the function h= 1r 2, where r is the distance function in M to S , satisfies the 2 followingcomparisontheorem: Theorem 7([10]). If M isaconnectedcompleteRiemannianmanifoldwithnon- positivesectionalcurvatureand S isa totallyconvex submanifoldofdimensiond thatisa closedsubsetof M,then foranyY T M, q / S , q ∈ ∈ (Hessh) (Y,Y) g¯(s (l),Y)2 q q′ ≥ wheres :[0,l] M istheuniqueunitgeodesicnormaltoS thatsatisfiess (0) q q S ands (l)=q→. ∈ q In[10]theconditionons isthatitmustsatisfyt =r (s (t)),butthisisequivalent tos meetsS orthogonally(see [9]chapter2). Proposition4. IfM isintheconditionsoftheorem7andF :M Misacomplete minimalimmersedsubmanifoldsuchthatforany p M F 1(S →), (s (l)) 2 ∈ \ − k F′(p) ⊤k ≥ 10