We give a characterization of the n-dimensional (n ≥ 3) hyperbolic cylinders in a Lorentzian space form. We show that the hyperbolic cylinders are the only complete space-like hypersurfaces in an (n + 1)-dimensional Lorentzian space form M₁ⁿ⁺¹(c) with non-zero constant mean curvature H whose two distinct principal curvatures λ and μ satisfy inf(λ - μ)² > 0 for c ≤ 0 or inf(λ - μ)² > 0, H² ≥ c, for c > 0, where λ is of multiplicity n - 1 and μ of multiplicity 1 and λ  0 при c ≤ 0 или inf(λ - μ)² > 0, H² ≥ c, при c > 0, где λ имеет порядок n - 1, а μ порядок 1 и λ < μ

Han, Annie Yi

Shu, Sh.

Zurnal matematiceskoj fiziki analiza geometrii

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Journal of Mathematical Physics, Analysis, Geometry2012, v. 8, No. 1, pp. 79–89Characterization of Hyperbolic Cylindersin a Lorentzian Space FormShichang ShuDepartment of Mathematics, Xianyang Normal UniversityXianyang 712000, Shaanxi, P. R. ChinaE-mail: shushichang@126.comAnnie Yi HanDepartment of Mathematics, Borough of Manhattan Community CollegeCUNY 10007 N. Y. USAE-mail: DrYHan@nyc.rr.comReceived January 8, 2009We give a characterization of the n-dimensional (n ≥ 3) hyperbolic cylin-ders in a Lorentzian space form. We show that the hyperbolic cylindersare the only complete space-like hypersurfaces in an (n + 1)-dimensionalLorentzian space form Mn+11 (c) with non-zero constant mean curvature Hwhose two distinct principal curvatures λ and µ satisfy inf(λ − µ)2 > 0 forc ≤ 0 or inf(λ− µ)2 > 0, H2 ≥ c, for c > 0, where λ is of multiplicity n− 1and µ of multiplicity 1 and λ < µ.Key words: space-like hypersurface, Lorentzian space form, mean curva-ture, principal curvature, hyperbolic cylinder.Mathematics Subject Classification 2000: 53C42, 53A10.1. IntroductionBy an (n + 1)-dimensional Lorentzian space form Mn+11 (c) we mean a Min-kowski space Rn+11 , a de Sitter space Sn+11 (c) or an anti-de Sitter space Hn+11 (c),according to c > 0, c = 0 or c < 0, respectively. That is, a Lorentzian space formMn+11 (c) is a complete simply connected (n+1)-dimensional Lorentzian manifoldwith constant curvature c. A hypersurface in a Lorentzian manifold is said to bespace-like if the induced metric on the hypersurface is positive definite.Project supported by NSF of Shaanxi Province (SJ08A31) and NSF of Shaanxi EducationalCommittee (11JK0479).c© Shichang Shu and Annie Yi Han, 2012Shichang Shu and Annie Yi HanIn connection with the negative settlement of the Bernstein problem due toCalabi [1] and Cheng-Yau [2], Choquet-Bruhat et al. [3] proved the followingtheorem:Theorem 1.1 ([3]). LetM be a complete space-like hypersurface in an (n+1)-dimensional Lorentzian space form Mn+11 (c), c ≥ 0. If M is maximal, then it istotally geodesic.T. Ishihara [4] also proved the following well-known result:Theorem 1.2 ([4]). Let M be an n-dimensional (n ≥ 2) complete maximalspace-like hypersurface in anti-de Sitter space Hn+11 (−1), thenS ≤ n, (1.1)and S = n if and only if M = Hm(− nm)×Hn−m(− nn−m),(1 ≤ m ≤ n− 1), whereS denotes the square of the norm of the second fundamental form of M .Recently, L. Cao and G. Wei [5] gave a new characterization of hyperboliccylinders in anti-de Sitter space Hn+11 (−1) as follows:Theorem 1.3 ([5]). Let M be an n-dimensional (n ≥ 3) complete maximalspace-like hypersurface with two distinct principal curvatures λ and µ in anti-deSitter space Hn+11 (−1). If inf(λ−µ)2 > 0, then M = Hm(− nm)×Hn−m(− nn−m),(1 ≤ m ≤ n− 1).As a generalization of Theorem 1.1, the complete space-like hypersurfaceswith constant mean curvature in a Lorentz manifold were studied by many math-ematicians, see, for instance, ([6–12]). We should note that two types of thewell-known standard models of complete space-like hypersurfaces with non-zeroconstant mean curvature in an (n+1)-dimensional Lorentzian space formMn+11 (c)are the totally umbilical space-like hypersurfaces and the following product man-ifolds:Hk(c1)× Sn−k(c2) in Sn+11 (c), (1c1+1c2=1c, c1 < 0, c2 > 0),Hk(c1)×Rn−k in Rn+11 , (c1 < 0, c = c2 = 0),Hk(c1)×Hn−k(c2) in Hn+11 (c), (1c1+1c2=1c, c1 < 0, c2 < 0),where k = 1, . . . , n−1. These three product hypersurfaces are respectively calledthe hyperbolic cylinders in Sn+11 (c), Rn+11 or Hn+11 (c). From U-H. Ki et al. [9], weknow that the hyperbolic cylinder H1(c1)×Sn−1(c2) in Sn+11 (c) has two distinctprincipal curvatures√c− c1 with multiplicity 1 and√c− c2 with multiplicityn− 1; the hyperbolic cylinder H1(c1)×Rn−1 in Rn+11 has two distinct principal80 Journal of Mathematical Physics, Analysis, Geometry, 2012, v. 8, No. 1Characterization of Hyperbolic Cylinders in a Lorentzian Space Formcurvatures√−c1 with multiplicity 1 and 0 with multiplicity n−1; the hyperboliccylinder H1(c1) × Hn−1(c2) in Hn+11 (c) has two distinct principal curvatures±√c− c1 with multiplicity 1 and ∓√c− c2 with multiplicity n− 1. The squareof the norm of the second fundamental form satisfies (1.3).U-H. Ki et al. [9] proved thatTheorem 1.4 ([9]). Let M be a complete space-like hypersurface with con-stant mean curvature in an (n+1)-dimensional Lorentzian space form Mn+11 (c).If one of the following properties holds(1) c ≤ 0,(2) c > 0, n ≥ 3 and n2H2 ≥ 4(n− 1)c,(3) c > 0, n = 2 and H2 > c,thenS ≤ −nc+ n3H22(n− 1) +n(n− 2)2(n− 1)√n2H4 − 4(n− 1)cH2, (1.2)where S denotes the square of the norm of the second fundamental form of M .As an application to Theorem 1.4, the authors of [9] gave a characterizationof hyperbolic cylinders of a Lorentzian space form Mn+11 (c) as follows:Theorem 1.5 ([9]). The hyperbolic cylinders are the only complete space-like hypersurfaces of Mn+11 (c) with non-zero mean curvature H and such that thesquare of the norm of the second fundamental form satisfiesS = −nc+ n3H22(n− 1) +n(n− 2)2(n− 1)√n2H4 − 4(n− 1)cH2. (1.3)About the same time, R. Aiyama [13] obtained a characterization of hyper-bolic cylinders of a Lorentzian 3-space form M31 (c) as follows:Theorem 1.6 ([13]). The hyperbolic cylinders are the only complete space-like surfaces in M31 (c) with non-zero constant mean curvature whose principalcurvatures λ and µ satisfy inf(λ− µ)2 > 0.It is natural for us to pose the following problem:Problem 1.1. Are the hyperbolic cylinders the only complete space-like hyper-surfaces in an (n+1)-dimensional Lorentzian space form Mn+11 (c) with non-zeroconstant mean curvature and two distinct principal curvatures λ and µ satisfyinginf(λ− µ)2 > 0?We should note that for c ≤ 0, L. Cao and G. Wei [5] posed the same problemas above, but they did not solve it. So the problem is still open.In this paper, we will give a characterization of the hyperbolic cylinders of(n+1)-dimensional Lorentzian space formMn+11 (c), which implies that the aboveJournal of Mathematical Physics, Analysis, Geometry, 2012, v. 8, No. 1 81Shichang Shu and Annie Yi HanProblem 1.1 can be solved affirmatively for c ≤ 0. For c > 0, we should note thatthe condition H2 ≥ c is necessary. We state our result as follows:Main Theorem. Let M be an n-dimensional (n ≥ 3) complete space-likehypersurface in an (n+1)-dimensional Lorentzian space form Mn+11 (c) with non-zero constant mean curvature and two distinct principal curvatures λ and µ ofmultiplicities n− 1 and 1 and λ < µ. Then:(1) For c = 0, if inf(λ− µ)2 > 0, then M is the hyperbolic cylinder H1(c1)×Rn−1, where c1 < 0;(2) For c < 0, if inf(λ− µ)2 > 0, then M is the hyperbolic cylinder H1(c1)×Hn−1(c2), where 1c1 +1c2= 1c , c1 < 0, c2 < 0;(3) For c > 0, if inf(λ−µ)2 > 0 and H2 ≥ c, thenM is the hyperbolic cylinderH1(c1)× Sn−1(c2), where 1c1 + 1c2 = 1c , c1 < 0, c2 > 0.R e m a r k 1.1. For the case n = 2, this Main Theorem was proved by R.Aiyama [13](see Theorem 1.6).2. PreliminariesLet M be an n-dimensional space-like hypersurface in an (n+1)-dimensionalLorentzian space form Mn+11 (c). We choose a local field of the semi-Riemannianorthonormal frames {e1, . . . , en+1} in Mn+11 (c) such that at each point of M ,{e1, . . . , en} span the tangent space of M and form an othonormal frame there.We use the following convention on the range of indices:1 ≤ A,B,C, . . . ,≤ n+ 1; 1 ≤ i, j, k, . . . ,≤ n.Let {ω1, . . . , ωn+1} be the dual frame field so that the semi-Riemannian metric ofMn+11 (c) is given by ds¯2 =∑iω2i −ω2n+1 =∑A²Aω2A, where ²i = 1 and ²n+1 = −1.The structure equations of Mn+11 (c) are given bydωA +∑B²BωAB ∧ ωB = 0, ωAB + ωBA = 0, (2.1)dωAB +∑C²CωAC ∧ ωCB = ΩAB, (2.2)whereΩAB = −12∑C,DKABCDωC ∧ ωD, (2.3)KABCD = ²A²Bc(δACδBD − δADδBC). (2.4)If we restrict these forms to M , we haveωn+1 = 0. (2.5)82 Journal of Mathematical Physics, Analysis, Geometry, 2012, v. 8, No. 1Characterization of Hyperbolic Cylinders in a Lorentzian Space FormCartan’s Lemma implies thatωn+1i =∑jhijωj , hij = hji. (2.6)The structure equations of M aredωi +∑jωij ∧ ωj = 0, ωij + ωji = 0, (2.7)dωij +∑kωik ∧ ωkj = −12∑k,lRijklωk ∧ ωl, (2.8)Rijkl = c(δikδjl − δilδjk)− (hikhjl − hilhjk), (2.9)where Rijkl are the components of the curvature tensor of M , andh =∑i,jhijωi ⊗ ωj (2.10)is the second fundamental form of M .From the above equation, we haven(n− 1)(R− c) = S − n2H2, (2.11)where n(n − 1)R is the scalar curvature of M,H is the mean curvature, andS =∑i,jh2ij is the square of the norm of the second fundamental form of M .The Codazzi equations arehijk = hikj , (2.12)where the covariant derivative of hij is defined by∑khijkωk = dhij −∑mhmjωmi −∑mhimωmj . (2.13)The second covariant derivative of hij is defined by∑lhijklωl = dhijk −∑mhmjkωmi −∑mhimkωmj −∑mhijmωmk. (2.14)Then we have the following Ricci identitieshijkl − hijlk =∑mhmjRmikl +∑mhimRmjkl. (2.15)Journal of Mathematical Physics, Analysis, Geometry, 2012, v. 8, No. 1 83Shichang Shu and Annie Yi HanIn a neighbourhood of a point x of M , we may choose orthonormal frame field{e1, . . . , en} such that hij = λiδij at x. We introduce the operator φ given by〈φX, Y 〉 = 〈hX, Y 〉 −H〈X,Y 〉. (2.16)Putting φ =∑i,jφijωi ⊗ ωj , where φij = hij − Hδij , we can easily see that φ istraceless, that the basis {e1, . . . , en} also diagonalizes φ at x with eigenvaluesµi = λi −H, and that|φ|2 =∑iµ2i =12n∑i,j(λi − λj)2 = S − nH2.Therefore, we know that |φ|2 ≡ 0 if and only if M is totally umbilical. We shallprove the following Lemma:Lemma 2.1. LetM be an n-dimensional space-like hypersurface in an (n+1)-dimensional Lorentzian space form Mn+11 (c) with constant mean curvature andtwo distinct principal curvatures λ and µ of multiplicities n−1 and 1 and λ < µ.Then12∆|φ|2 = |∇φ|2 + |φ|2{|φ|2 − n(n− 2)H√n(n− 1) |φ|+ n(c−H2)}. (2.17)P r o o f. We firstly need the following result due to [14] and [15] : Letµ1, µ2, . . . , µn be real numbers such that∑iµi = 0 and∑iµ2i = β2, where β =const ≥ 0, then− n− 2√n(n− 1)β3 ≤∑iµ3i ≤n− 2√n(n− 1)β3, (2.18)and equality holds in the right-hand(left-hand) side if and only if (n − 1) of theµ′is are non-positive and equal ((n− 1) of the µ′is are non-negative and equal).Now we put µi = λi − H, then∑iµi = 0 and∑iµ2i = |φ|2. Since M hastwo distinct principal curvatures λ and µ of multiplicities n − 1 and 1, withoutloss of generality, we may assume λ1 = · · · = λn−1 = λ, µ = λn, where λi fori = 1, 2, . . . , n are the principal curvatures of M . Therefore, we know that(n− 1)λ+ µ = nH, S = (n− 1)λ2 + µ2. (2.19)Since we assume that λ < µ, from (2.19), we have n(λ − H) = λ − µ < 0.Therefore, we know that µ1 = · · · = µn−1 = λ − H < 0. We infer that theequality holds in the right-hand side of (2.18), that is,∑iµ3i =n− 2√n(n− 1) |φ|3. (2.20)84 Journal of Mathematical Physics, Analysis, Geometry, 2012, v. 8, No. 1Characterization of Hyperbolic Cylinders in a Lorentzian Space FormFrom [8] or [10], we have the well-known Simons’ formula of Lorentzian versionas follows12∆|φ|2 = |∇φ|2 + (|φ|2)2 − nHtrφ3 + n(c−H2)|φ|2. (2.21)From (2.20) and (2.21), we see that Lemma 2.1 is true.The following generalized maximum principle will be important in the sequel.Proposition 2.1 ([16, 17]). Let M be a complete Riemannian manifold withRicci curvature bounded from below and f a C2-function which is bounded frombelow on M . Then there is a point sequence xk in M such thatlimk→∞f(xk) = inf(f), limk→∞|∇f(xk)| = 0, limk→∞inf ∆f(xk) ≥ 0.Now we state a proposition which can be proved by making use of the similarmethod due to Otsuki [18].Proposition 2.2. Let M be a hypersurface in an (n + 1)-dimensionalLorentzian space form Mn+11 (c) such that the multiplicities of the principal cur-vatures are constant. Then the distribution of the space of the principal vectorscorresponding to each principal curvature is completely integrable. In particular,if the multiplicity of a principal curvature is greater than 1, then this principalcurvature is constant on each integral submanifold of the corresponding distribu-tion of the space of the principal vectors.3. Proof of Main TheoremWe denote the integral submanifold through x ∈ Mn corresponding to λ byMn−11 (x). Puttingdλ =n∑k=1λ,k ωk, dµ =n∑k=1µ,k ωk. (3.1)From Proposition 2.2, we haveλ,1= λ,2= · · · = λ,n−1= 0 on Mn−11 (x). (3.2)From (2.19), we havedµ = −(n− 1)dλ. (3.3)Hence, we also haveµ,1= µ,2= · · · = µ,n−1= 0 on Mn−11 (x). (3.4)Journal of Mathematical Physics, Analysis, Geometry, 2012, v. 8, No. 1 85Shichang Shu and Annie Yi HanFrom (2.13), we have ∑khijkωk = dλiδij + (λj − λi)ωij . (3.5)We infer thathijk = 0, for any k, if i 6= j, 1 ≤ i ≤ n− 1 and 1 ≤ j ≤ n− 1. (3.6)From (3.1), (3.2) and (3.5), we have for 1 ≤ j ≤ n− 1,dλ = dλj =n∑k=1hjjkωk=n−1∑k=1hjjkωk + hjjnωn = λ,n ωn.(3.7)Therefore, we have for 1 ≤ j ≤ n− 1,hjjk = 0, 1 ≤ k ≤ n− 1, and hjjn = λ,n . (3.8)From (3.1), (3.4) and (3.5), we havedµ = dλn =n∑k=1hnnkωk=n−1∑k=1hnnkωk + hnnnωn =n∑i=1µ,i ωi = µ,n ωn.(3.9)Hence, we obtainhnnk = 0, 1 ≤ k ≤ n− 1, and hnnn = µ,n . (3.10)Now we prove the following Lemma:Lemma 3.1. LetM be an n-dimensional space-like hypersurface in an (n+1)-dimensional Lorentzian space form Mn+11 (c) with a constant mean curvature andtwo distinct principal curvatures λ and µ of multiplicities n− 1 and 1. Then|∇|φ|2|2 = 4n|φ|2n+ 2|∇φ|2, (3.11)where φ is defined by (2.16).P r o o f. From (2.19), we have|φ|2 = S − nH2 = n(n− 1)λ2 − 2n(n− 1)λH + n(n− 1)H2 (3.12)= n(n− 1)(λ−H)2.86 Journal of Mathematical Physics, Analysis, Geometry, 2012, v. 8, No. 1Characterization of Hyperbolic Cylinders in a Lorentzian Space FormHence, from (3.2) we obtain|∇|φ|2|2 =∑k(|φ|2,k)2 =∑k[2n(n− 1)(λ−H)λ,k]2 (3.13)= 4n2(n− 1)2(λ−H)2(λ,n)2.Since φij = hij −Hδij , from (3.3), (3.6), (3.8) and (3.10), we have|∇φ|2 = |∇h|2 =∑i,j,kh2ijk =n−1∑i,j,k=1h2ijk + 3n−1∑i,j=1h2ijn + 3n−1∑i=1h2inn + h2nnn (3.14)= 3n−1∑i=1h2iin + h2nnn = 3(n− 1)(λ,n)2 + (µ,n)2= 3(n− 1)(λ,n)2 + (n− 1)2(λ,n)2 = (n− 1)(n+ 2)(λ,n)2.From (3.12), (3.13) and (3.14), we have|∇|φ|2|2 = 4n2(n− 1)2(λ−H)2 |∇φ|2(n− 1)(n+ 2)=4n2(n− 1)(λ−H)2n+ 2|∇φ|2 = 4n|φ|2n+ 2|∇φ|2.So, the proof of Lemma 3.1 is completed.P r o o f of Main Theorem. Since we assume that inf(λ− µ)2 > 0, we have(λ−µ)2 > 0. Putting (λ−µ)2 = κ > 0, we have [n(λ−H)]2 = (λ−µ)2 = κ > 0.Therefore, we know that|φ|2 = n(n− 1)(λ−H)2 = n− 1nκ > 0,that is, M is not umbilical. From Lemma 2.1 and Lemma 3.1, we have12∆|φ|2 = n+ 24n|φ|2 |∇|φ|2|2 + |φ|2{|φ|2 − n(n− 2)H√n(n− 1) |φ|+ n(c−H2)}. (3.15)Since the Ricci curvature Rii ≥ (n− 1)c− n2H24 and |φ|2 = n(n− 1)(λ−H)2 =n−1n κ > 0 are bounded from below, from Proposition 2.1, we have that there is apoint sequence xk in M such thatlimk→∞|φ|2(xk) = inf(|φ|2), limk→∞|∇|φ|2(xk)| = 0, limk→∞inf ∆|φ|2(xk) ≥ 0.By (3.15), we haveinf |φ|2{inf |φ|2 − n(n− 2)H√n(n− 1) inf |φ|+ n(c−H2)} ≥ 0.Journal of Mathematical Physics, Analysis, Geometry, 2012, v. 8, No. 1 87Shichang Shu and Annie Yi HanSince inf |φ|2 > 0, we haveinf |φ|2 − n(n− 2)H√n(n− 1) inf |φ|+ n(c−H2) ≥ 0. (3.16)Since for c > 0, H2 ≥ c implies n2H2 ≥ 4(n−1)c, we know that the discriminantof (3.16) is non-negative for all c. From (3.16), we haveinf |φ| ≤ 12√nn− 1[(n− 2)H −√n2H2 − 4(n− 1)c], (3.17)orinf |φ| ≥ 12√nn− 1[(n− 2)H +√n2H2 − 4(n− 1)c]. (3.18)Assume that (3.17) holds, if c ≤ 0, we have inf |φ| ≤ 12√nn−1 [(n−2)H−nH] < 0,which contradicts inf |φ|2 > 0; if c > 0, since we assume that H2 ≥ c, we haveinf |φ| ≤ 12√nn−1 [(n−2)H−√n2H2 − 4(n− 1)c] ≤ 0, this is also in contradictionto inf |φ|2 > 0. Therefore, we know that (3.18) holds, we have|φ|2 ≥ 14nn− 1[(n− 2)H +√n2H2 − 4(n− 1)c]2,and this is equivalent toS ≥ −nc+ n3H22(n− 1) +n(n− 2)2(n− 1)√n2H4 − 4(n− 1)cH2.From Theorem 1.4, we haveS = −nc+ n3H22(n− 1) +n(n− 2)2(n− 1)√n2H4 − 4(n− 1)cH2.By Theorem 1.5, we see that Main Theorem is true.Acknowledgment. The authors would like to thank the referee for his/hermany valuable suggestions and comments made that significantly improved thepaper.References[1] E. Calabi, Examples of Bernstein Problems for Some Nonlinear Equations. — Proc.Symp. Pure Appl. Math. 15 (1970), 223–230.[2] S.Y. Cheng and S.T. Yau, Maximal Space-Like Hypersurfaces in the Lorentz–Minkowski Spaces. — Ann. 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Characterization of Hyperbolic Cylinders in a Lorentzian Space Form

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Characterization of Hyperbolic Cylinders in a Lorentzian Space Form

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