Shu, Sh.

Zurnal matematiceskoj fiziki analiza geometrii

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

Journal of Mathematical Physics, Analysis, Geometry2008, vol. 4, No. 2, pp. 294304Complete Hypersurfaces in a Real Space FormShu ShichangDepartment of Mathematics, Xianyang Teachers UniversityXianyang, 712000, Shaanxi, Peoples Republic of ChinaE-mail:shushichang@126.comReceived February 24, 2007Let Mnbe an n-dimensional complete hypersurface with the scalar cur-vature n(n   1)R and the mean curvature H being linearly related, thatis, n(n   1)R = k0H(k0> 0) in a real space form Rn+1(c). Assume thatthe mean curvature is positive and obtains its maximum on Mn. We showthat (1) if c = 1; k0 2npn(n  1), for any i,Pj 6=i2j> n(n   1) andjhj2 nH2+ (B+H)2, then Mnis totally umbilical, or (i) n  3;Mnis lo-cally an H(r)-torus with r2<n 1n, (ii) n = 2;Mnis locally an H(r)-toruswith r26=n 1n; (2) if c = 0 and jhj2 nH2+(eB+H)2, thenMnis isometric toa standard round sphere, a hyperplane Rnor Sn 1(c1)  R1; (3) if c =  1and jhj2 nH2+ (bB+H)2, then Mnis totally umbilical or is isometric toSn 1(r) H1( 1=(r2+ 1)) for some r > 0, where jhj2denotes the squarednorm of the second fundamental form of Mn; B+H;eB+HandbB+Hare denotedby (1.1), (1.2) and (1.3).Key words: hypersurface, mean curvature, scalar curvature, real spaceform.Mathematics Subject Classication 2000: 53C40 (primary); 53C20(secondary).1. IntroductionLet Rn+1(c) be an (n+ 1)-dimensional connected Riemannian manifold withconstant sectional curvature c. We also call it a real space form. According toc > 0, c = 0 or c < 0, it is called sphere space, Euclidean space or hyperbolicspace, respectively, and it is denoted by Sn+1(c); Rn+1and Hn+1(c). As it is well-known that there are many rigidity results for hypersurfaces with constant meancurvature or with constant scalar curvature in Sn+1(c); Rn+1and Hn+1(c), forexample, see [13, 5] and [12] etc., but fewer ones are obtained for hypersurfacesThis work is supported in part by the Natural Science Foundation of China and NSF ofShaanxi, China.c Shu Shichang, 2008Complete Hypersurfaces in a Real Space Formwith the scalar curvature and the mean curvature being linearly related. We knowthat an H(r)-torus in a unit sphere Sn+1(1) is the product immersion Sn 1(r)S1(p1  r2) ,! RnR2, where Sn 1(r)  Rn; S1(p1  r2)  R2, 0 < r < 1, arestandard immersions. In some orientation, H(r)-torus has principal curvaturesgiven by 1=    = n 1=p1 r2rand n=  rp1 r2.In [12], the authors obtained the following:Theorem 1.1. Let Mnbe an n-dimensional complete hypersurface with con-stant mean curvature H in a unit sphere Sn+1(1). (1) If jhj2< D0(n;H), thenMnis totally umbilical. (2) If jhj2= D0(n;H), then (i) when H = 0;Mnislocally a Cliord torus; (ii) when H 6= 0; n  3;Mnis locally an H(r)-torus withr2<n 1n; (iii) when H 6= 0; n = 2;Mnis locally an H(r)-torus with r26=n 1n,whereD0(n;H) = n+n3H22(n  1) (n  2)nH2(n  1)[n2H2+ 4(n  1)]12:In [6], S.Y. Cheng and S.T. Yau obtained the following:Theorem 1.2. Let Mnbe a complete hypersurface with constant mean cur-vature in Rn+1. If the sectional curvature of Mnis nonnegative, then Mnisisometric to a standard round sphere, a hyperplane Rnor a Riemannian productSk(c1)Rn k, 1  k  n  1.In this paper, we study the hypersurfaces in a real space form Rn+1(c) withscalar curvature n(n   1)R and the mean curvature H being linearly related.We obtain the following theorems:Theorem 1.3. Let Mnbe an n-dimensional complete hypersurface with n(n 1)R = k0H in a unit sphere Sn+1(1), where k0( 2npn(n  1)) is a positiveconstant. Assume that the mean curvature H is positive and obtains its maximumon Mnand for any i,Pj 6=i2j> n(n  1), where j(j = 1; : : : ; i  1; i+1; : : : ; n)are the principal curvatures onMn. If the squared norm of the second fundamentalform jhj2satisesjhj2 nH2+ (B+H)2on Mn, then Mnis totally umbilical, or (i) n  3;Mnis locally an H(r)-toruswith r2<n 1n; (ii) n = 2;Mnis locally an H(r)-torus with r26=n 1n, whereB+H=  12(n  2)rnn  1H +sn3H24(n  1)+ n: (1:1)Theorem 1.4. Let Mnbe an n-dimensional complete hypersurface with n(n 1)R = k0H in a Euclidean space Rn+1, where k0is a positive constant. AssumeJournal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 2 295Shu Shichangthat the mean curvature H is positive and obtains its maximum on Mn. If thesquared norm of the second fundamental form jhj2satisesjhj2 nH2+ (eB+H)2on Mn, then Mnis isometric to a standard round sphere, a hyperplane Rnora Riemannian product Sn 1(c1)R1, whereeB+H=rnn  1H: (1:2)Theorem 1.5. Let Mnbe an n-dimensional complete hypersurface with n(n 1)R = k0H in a hyperbolic space Hn+1( 1), where k0is a positive constant.Assume that the mean curvature H is positive and obtains its maximum on Mn.If the squared norm of the second fundamental form jhj2satisesjhj2 nH2+ (bB+H)2on Mn, then Mnis totally umbilical or is isometric to Sn 1(r)H1( 1=(r2+1))for some r > 0, wherebB+H=  12(n  2)rnn  1H +sn3H24(n  1)  n; (n2H2 4(n  1)): (1:3)2. PreliminariesLet Mnbe an n-dimensional hypersurface in Rn+1(c). For any p 2 Mnwe choose a local orthonormal frame e1; : : : ; en; en+1in Rn+1(c) around p suchthat e1; : : : ; enare tangential to Mn. Take the corresponding dual co-framef!1; : : : ; !n; !n+1g. In this paper we make the following convention on the rangeof indices,1  A;B;C     n+ 1; 1  i; j; k;     n:The structure equations of Rn+1(c) ared!A=PB!AB^ !B; !AB=  !BA;d!AB=Pc!AC^ !CB  c!A^ !B:If we denote by the same letters the restrictions of !A; !ABto Mn, we haved!i=Xj!ij^ !j; !ij=  !ji; (2:1)296 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 2Complete Hypersurfaces in a Real Space Formd!ij=Xk!ik^ !kj 12Xk;lRijkl!k^ !l; (2:2)where Rijklis the curvature tensor of the induced metric on Mn.Restricted to Mn; !n+1= 0, thus0 = d!n+1=Xi!n+1i^ !i; (2:3)and by Cartan's lemma we can write!in+1=Xjhij!j; hij= hji: (2:4)The quadratic form h =Pi;jhij!i!jis the second fundamental form of Mn.The Gauss equation isRijkl= c(ÆikÆjl  ÆilÆjk) + hikhjl  hilhjk; (2:5)n(n  1)R = n(n  1)c+ n2H2  jhj2; (2:6)where R is the normalized scalar curvature, H = (1=n)Pihiithe mean curvatureand jhj2=Pi;jh2ijthe squared norm of the second fundamental form of Mn,respectively.The Codazzi equation and Ricci identity arehijk= hikj; (2:7)hijkl  hijlk=XmhmjRmikl+XmhimRmjkl; (2:8)where the rst and the second covariant derivatives of the second fundamentalform are dened byXkhijk!k= dhij+Xkhkj!ki+Xkhik!kj; (2:9)Xlhijkl!l= dhijk+Xmhmjk!mi+Xmhimk!mj+Xmhijm!mk: (2:10)In order to represent our theorems, we need some notations, for details seeH.B. Lawson [9] and P.J. Ryan [11]. First we give a description of the real hyper-bolic space Hn+1(c) of constant curvature c(< 0).For any two vectors x and y in Rn+2, we setg(x; y) = x1y1+ : : :+ xn+1yn+1  xn+2yn+2;Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 2 297Shu Shichang(Rn+2; g) is the so-called Minkowski space-time. Denote  =p 1=c. We deneHn+1(c) = fx 2 Rn+2j g(x; x) =  2; xn+2> 0g:Then Hn+1(c) is a simply-connected hypersurface of Rn+2. Hence, we obtaina model of a real hyperbolic space.We deneM1= fx 2 Hn+1(c) j x1= 0g;M2= fx 2 Hn+1(c) j x1= r > 0g;M3= fx 2 Hn+1(c) j xn+2= xn+1+ g;M4= fx 2 Hn+1(c) j x21+ : : :+ x2n+1= r2> 0g;M5= fx 2 Hn+1(c) j x21+ : : :+ x2k+1= r2> 0;x2k+2+ : : :+ x2n+1  x2n+2=  2  r2g:M1; : : : ;M5are often called the standard examples of complete hypersurfaces inHn+1(c) with at most two distinct constant principal curvatures. It is obviousthat M1; : : : ;M4are totally umbilical. In the sense of Chen [7], they are calledthe hyperspheres of Hn+1(c). M3is called the horosphere and M4 the geodesicdistance sphere of Hn+1(c). P.J. Ryan [11] obtained the following:Lemma 2.1([11]). Let Mnbe a complete hypersurface in Hn+1(c). Supposethat, under a suitable choice of a local orthonormal tangent frame eld of TMn,the shape operator over TMnis expressed as a matrix A. If Mnhas at most twodistinct constant principal curvatures, then it is congruent to one of the following:(1) M1. In this case, A = 0, andM1is totally geodesic. Hence M1is isometricto Hn(c).(2) M2. In this case, A =1=2p1=2+1=r2In, where Indenotes the identity matrixof degree n, and M2is isometric to Hn( 1=(r2+ 2)).(3) M3. In this case, A =1In, and M3is isometric to a Euclidean space Rn.(4) M4. In this case, A =p1=r2+ 1=2In;M4is isometric to a round sphereSn(r) of radius r.(5) M5. In this case, A = Ik In k, where  =p1=2+ 1=r2, and =1=2p1=r2+1=2;M5is isometric to Sk(r)Hn k( 1=(r2+ 2)).We also need the following algebraic Lemma due to [10] and [1].Lemma 2.2([10, 1]). Let i; i = 1; : : : ; n be real numbers, withPii= 0andPi2i= 2 0. Then n  2pn(n  1)3Xi3in  2pn(n  1)3; (2:11)298 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 2Complete Hypersurfaces in a Real Space Formand equality holds if and only if either (n   1) of the numbers iare equal to=pn(n  1) or (n  1) of the numbers iare equal to  =pn(n  1).3. Proof of TheoremsIn order to prove our theorems, we introduce an operator 2 due to S.Y. Chengand S.T. Yau [5] by2f =Xi;j(nHÆij  hij)fij; (3:1)where f is a C2-function on Mn, the gradient and Hessian (fij) are dened bydf =Xifi!i;Xjfij!j= dfi+Xjfj!ji: (3:2)The Laplacian of f is dened by f =Pifii.We choose a local frame eld e1; : : : ; enat each point of Mn, such that hij=iÆij. From (3.1) and (2.6), we have2(nH) = nH(nH) Pii(nH)ii=12(nH)2 Pi(nH)2i Pii(nH)ii=12n(n  1)R+12jhj2  n2jrHj2 Pii(nH)ii: (3.3)From (2.7) and (2.8), by a standard and direct calculation, we have12jhj2=Xi;j;kh2ijk+Xii(nH)ii+12Xi;jRijij(i  j)2; (3:4)where Rijij= c + ij(i 6= j) denotes the sectional curvature of the sectionspanned by fei; ejg:From (3.3) and (3.4), we get2(nH) =12n(n  1)R+ jrhj2  n2jrHj2+12Xi;j(c+ ij)(i  j)2: (3:5)By making use of the similar method in [4], we can prove the following:Proposition 3.1. Let Mnbe an n-dimensional hypersurface in a real spaceform Rn+1(c) with n(n  1)R = k0H, k0= constant > 0. Assume that the meancurvature H > 0. Then we have the operatorL = 2  (k0=2n)Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 2 299Shu Shichang(1) if c > 0 and for any i,Pj 6=i2j> n(n  1)c; L is elliptic;(2) if c  0; L is elliptic.P r o o f. We choose an orthonormal frame eld fejg at each point in Mnsothat hij= iÆij. For any i,(nH   i  k0=2n) =Pjj  i  (1=2)[ Pj2j+ n2H2+ n(n  1)c]=(nH)= [(Pjj)2  iPjj  (1=2)Pl 6=jlj  (1=2)n(n  1)c](nH) 1= [Pj2j+ (1=2)Pl 6=jlj  iPjj  (1=2)n(n   1)c](nH) 1= [Pi6=j2j+ (1=2)Pl6=jl;j 6=ilj  (1=2)n(n   1)c](nH) 1= (1=2)[Pj 6=i2j+ (Pj 6=ij)2  n(n  1)c](nH) 1: (3.6)(1) If c > 0 and for any i,Pj 6=i2j> n(n  1)c; from (3.6), we have(nH   i  k0=2n)  (1=2)[Xj 6=i2j  n(n  1)c](nH) 1> 0:Therefore, we know that L is an elliptic operator.(2) If c  0, from (3.6) again, we have(nH   i  k0=2n) > 0:Therefore, we also know that L is an elliptic operator. This completes theproof of Prop. 3.1.We can also prove the following:Proposition 3.2. Let Mnbe an n-dimensional hypersurface in a real spaceform Rn+1(c) with n(n  1)R = k0H, k0= constant > 0. Assume that the meancurvature H > 0. Then we have:(1) if c > 0 and k0 2npn(n  1)c, thenjrhj2 n2jrHj2;(2) if c  0, for all k0> 0, thenjrhj2 n2jrHj2:300 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 2Complete Hypersurfaces in a Real Space FormP r o o f. Since H > 0, we have jhj26= 0. In fact, if jhj2=Pi2i= 0 ata point of Mn, then i= 0, i = 1; 2; : : : ; n, at this point. Therefore H = 0 at thispoint. This is impossible.From (2.6) and n(n  1)R = k0H, we havek0riH = 2n2HriH   2Pj;khkjhkji;(12k0  n2H)riH =  Pj;khkjhkji;(12k0  n2H)2jrHj2=Pi(Pj;khkjhkji)2Pi;jh2ijPi;j;kh2ijk= jhj2jrhj2:Therefore, we havejrhj2  n2jrHj2 [(k02  n2H)2  n2jhj2]jrHj21jhj2= [(k0)24  n3(n  1)c]jrHj21jhj2: (3.7)(1)If c > 0 and k0 2npn(n  1)c, from (3.7), we havejrhj2  n2jrHj2 0:(2) If c  0, from (3.7), we also havejrhj2  n2jrHj2 0:This completes the proof of Prop. 3.2.Proposition 3.3. Let Mnbe an n-dimensional hypersurface in a real spaceform Rn+1(c) with n(n  1)R = k0H, k0= constant > 0. Then we havenLH  (jrhj2  n2jrHj2) + jgj2fnc+ nH2 n(n  2)pn(n  1)jHjjgj   jgj2g;where jgj2is a nonnegative C2-function on Mndened by jgj2= jhj2  nH2.P r o o f. From (3.5) we havenLH = n[2H   (k0=2n)H]= 2(nH)  (1=2)[n(n  1)R]= jrhj2  n2jrHj2+12Pi;j(c+ ij)(i  j)2= jrhj2  n2jrHj2+ ncjhj2  n2H2c  jhj4+ nHPi3i: (3.8)Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 2 301Shu ShichangLet jgj2be a nonnegative C2-function on Mndened by jgj2= jhj2  nH2.SincePi(H   i) = 0,Pi(H   i)2= jhj2  nH2= jgj2, by Lem. 2.2 we getnHPi3i= 3nH2jhj2  2n2H4  nHPi(H   i)3 3nH2jgj2+ n2H4  njHjn  2pn(n  1)jgj3: (3.9)Therefore, from (3.8) and (3.9), we havenLH  jrhj2  n2jrHj2+ jgj2fnc+ nH2 n(n  2)pn(n  1)jHjjgj   jgj2g:This completes the proof of Prop. 3.3.P r o o f o f T h e o r e m 1.3. From the assumption of Th. 1.3, Prop. 3.2and Prop. 3.3 for c = 1, we havenLH  jgj2fn+ nH2 n(n 2)pn(n 1)Hjgj   jgj2g = jgj2PH(jgj);(3:10)wherePH(jgj) = n+ nH2 n(n  2)pn(n  1)Hjgj   jgj2:PH(jgj) has two real roots B Hand B+Hgiven byBH=  12(n  2)rnn  1H sn3H24(n  1)+ n:Therefore, we know thatPH(jgj) = (jgj  B H)( jgj +B+H):Clearly, we know that jgj B H> 0. From the assumption of Th. 1.3, we infer thatPH(jgj)  0 onMn. This implies that the right-hand side of (3.10) is nonnegative.Since, from Prop. 3.1, the operator L is elliptic, and H obtains its maximum onMn, from (3.10) we know that H = const on Mn. Therefore, we know that Mnis an n-dimensional complete hypersurface with constant mean curvature H(> 0)in a unit sphere Sn+1(1). By the assumption of Th. 1.3 and the result of Th. 1.1,we can check directly that jhj2 nH2+ (B+H)2= n+n3H22(n 1) (n 2)nH2(n 1)[n2H2+4(n   1)]12= D0(n;H). Therefore we have either Mnis totally umbilical, or (i)n  3;Mnis locally an H(r)-torus with r2<n 1n; (ii) n = 2;Mnis locally anH(r)-torus with r26=n 1n. This completes the proof of Th. 1.3.302 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 2Complete Hypersurfaces in a Real Space FormP r o o f o f T h e o r e m 1.4. From the assumption of Th. 1.4, Prop. 3.2and Prop. 3.3, for c = 0, we havenLH  jgj2fnH2 n(n 2)pn(n 1)Hjgj   jgj2g = jgj2QH(jgj);(3:11)whereQH(jgj) = nH2 n(n  2)pn(n  1)Hjgj   jgj2:QH(jgj) has two real rootseB HandeB+Hgiven byeB H=  (n  1)rnn  1H;eB+H=rnn  1H:Therefore, we know thatQH(jgj) = (jgj  eB H)( jgj +eB+H):Clearly, we know that jgj  eB H> 0. From the assumption of Th. 1.4, we inferthat QH(jgj)  0 on Mn. This implies that the right-hand side of (3.11) isnonnegative. From Proposition 3.1, we know that L is elliptic, and H obtains itsmaximum on Mn. From (3.11), we have H = const on Mn. From (3.11) again,we get jgj2QH(jgj) = 0. We infer that the equality holds in Lem. 2.2. Therefore,we know that (n   1) of the numbers H   iare equal to jgj=pn(n  1). Thisimplies that Mnhas (n   1) principal curvatures equal and constant. As H isconstant, the other principal curvature is constant as well. From an inequality ofChenOkumura [8], we know that jhj2 n2H2=(n  1) implies that the sectionalcurvature K of Mnis nonnegative. Therefore, we know that Mnis a completehypersurface in Rn+1with constant mean curvature and nonnegative sectionalcurvature. From Theorem 1.2, we have either Mnis isometric to a standardround sphere, a hyperplane Rnor a Riemannian product Sn 1(c1)  R1. Thiscompletes the proof of Th. 1.4.P r o o f o f T h e o r e m 1.5. From the assumption of Th. 1.5, Prop. 3.2and Prop. 3.3, for c =  1, we havenLH  jgj2f n+ nH2 n(n 2)pn(n 1)Hjgj   jgj2g = jgj2RH(jgj);(3:12)whereRH(jgj) =  n+ nH2 n(n  2)pn(n  1)Hjgj   jgj2:RH(jgj) has two real rootsbB HandbB+Hgiven bybBH=  12(n  2)rnn  1H sn3H24(n  1)  n; n2H2 4(n  1):Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 2 303Shu ShichangTherefore, we know thatRH(jgj) = (jgj  bB H)( jgj +bB+H):Clearly, we know that jgj bB H> 0. From the assumption of Th. 1.5, we infer thatRH(jgj)  0 onMn. This implies that the right-hand side of (3.12) is nonnegative.From Proposition 3.1, we know that L is elliptic. Since H obtains its maximumon Mn, from (3.12), we have H = const on Mn. From (3.12) again, we getjgj2RH(jgj) = 0, so jgj2= 0, and Mnis totally umbilical, or RH(jgj) = 0. In thelatter case, we know that (n 1) of the numbers H iare equal to jgj=pn(n  1).This implies thatMnhas (n 1) principal curvatures equal and constant. As H isconstant, the other principal curvature is constant as well, soMnis isoparametric.From the result of Lem. 2.1, Mnis isometric to Sn 1(r) H1( 1=(r2+ 1)) forsome r > 0. This completes the proof of Th. 1.5.References[1] H. Alencar and M.P. do Carmo, Hypersurfaces with Constant Mean Curvature inSphere.  Proc. Amer. Math. Soc. 120 (1994), 12231229.[2] Q.M. Cheng, Complete Hypersurfaces in Euclidean Space Rn+1with ConstantScalar Curvature.  Indiana Univ. Math. J. 51 (2002), 5368.[3] Q.M. Cheng, Hypersurfaces in a Unit Shere Sn+1(1) with Constant Scalar Curva-ture.  J. London Math. Soc. 64 (2001), 755768.[4] Q.M. Cheng, Complete Space-Like Hypersurfaces of a de Sitter Space with r = kH . Mem. Fac. Sci. Kyushu Univ. 44 (1990), 6777.[5] S.Y. Cheng and S.T. Yau, Hypersurfaces with Constant Scalar Curvature.  Math.Ann. 225 (1977), 195204.[6] S.Y. Cheng and S.T. Yau, Dierential Equations on Riemannian Manifolds andtheir Geometric Applications.  Comm. Pure Appl. Math. 28 (1975), 333354.[7] B.Y. Chen, Totally Mean Curvature and Submanifolds of Finite Type. World Sci.,Singapore, 1984.[8] B.Y. Chen and M. Okumura, Scalar Curvature, Inequality and Submanifold. Proc. Amer. Math. Soc. 38 (1973), 605608.[9] H.B. Lawson, Jr., Local Rigidity Theorems for Minimal Hypersurfaces.  Ann.Math. 89(2) (1969), 187197.[10] M. Okumura, Hypersurfaces and a Pinching Problem on the Second FundamentalTensor.  Amer. J. Math. 96 (1974), 207213.[11] P.J. Ryan, Hypersurfaces with Parallel Ricci Tensor.  Osaka J. Math. 8 (1971),251259.[12] S.C. Shu, Complete Hypersurfaces with Constant Mean Curvature in Locally Sym-metric Manifold.  Adv. Math. Chinese 33 (2004), 563569.304 Journal of Mathematical Physics, Analysis, Geometry, 2008, vol. 4, No. 2

Complete Hypersurfaces in a Real Space Form

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Complete Hypersurfaces in a Real Space Form

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