A CHARACTERIZATION OF THE CLIFFORD TORUS

Transcription

1 PROCEEDINGS OF THE AERICAN ATHEATICAL SOCIETY Volue 17, Nuber 3, arch 1999, Pages S (99) A CHARACTERIZATION OF THE CLIFFORD TORUS QING-ING CHENG AND SUSUU ISHIKAWA (Coucated by Chrstopher Croke) Abstract. I ths paper, we prove that a -desoal closed al hypersurface wth Rcc curvature Rc() of a ut sphere S+1 (1) s soetrc to a Clfford torus f S + 14(+4),whereSs the squared 9+30 or of the secod fudaetal for of. 1. Itroducto Let be a -desoal closed al hypersurface a ut sphere S +1 (1) of deso + 1. LetS deote the squared or of the secod fudaetal for of. Fro the Gauss equato (see secto ), we kow that S, whch s extrsc by defto, s actually a trsc quatty. It s well-kow that Cher, do Caro ad Kobayash [3] ad Lawso [4] obtaed depedetly that Clfford tor are the oly closed al hypersurfaces of the ut sphere wth S =. Whe the scalar curvature of s costat, Yag ad the frst aed author proved [6] ad [7] that f S + 3,thes soetrc to a Clfford torus S ( ) S ( ). A atural proble s that, for a closed al hypersurface of a ut sphere, whether there exsts a costat ɛ() > 0 such that f S + ɛ(), the S = ad s soetrc to a Clfford torus S ( ) S ( ). The frst aed author [] gave a postve aswer uder the addtoal codto that has oly two dstct prcpal curvatures. I geeral, t stll reas ope ad t s a very hard proble. O the other had, the Clfford torus S ( ) S ( ) s a closed al hypersurface S +1 (1) wth S = ad ts Rcc curvature vares betwee ( 1) ( 1). If, the Rc(). Hece t s atural to ask ad Receved by the edtors ay 15, 1996 ad, revsed for, Noveber 1, atheatcs Subject Classfcato. Prary 53C0, 53C4. Key words ad phrases. al hypersurfaces, scalar curvature, Rcc curvature, Clfford torus. The frst author s research was partally supported by a Grat--Ad for Scetfc Research fro the Japaese stry of Educato, Scece ad Culture ad by a Grat--Ad for Scetfc Research fro Josa Uversty. The secod author s research was partally supported by a Grat--Ad for Scetfc Research fro the Japaese stry of Educato, Scece ad Culture. 819 c 1999 Aerca atheatcal Socety Lcese or copyrght restrctos ay apply to redstrbuto; see

2 80 QING-ING CHENG AND SUSUU ISHIKAWA whether there exsts a costat ɛ() > 0 such that f s a closed al hypersurface wth Rc() ad S + ɛ(), the S = ad s soetrc to a Clfford torus S ( ) S ( )(1<< 1). I ths paper, we gve a affratve aswer for the above proble. Theore 1. Let be a -desoal closed al hypersurface of a ut sphere S +1 (1) wth Rcc curvature Rc().If 14( +4) 9+30, S + the S = ad s soetrc to a Clfford torus S ( ) S ( (1 << 1). I partcular, f 5, we obta the followg ) Theore. Let be a -desoal ( 5) closed al hypersurface of a ut sphere S +1 (1). If S +ɛ(), the S = ad s soetrc to a Clfford torus S ( ) S ( ), where ɛ(3) = 4 85, ɛ(4) = 8 3( ad ɛ(5) = Reark. For 5, Peg-Terg [5] proved the followg: Let be a -desoal ( 5) closed al hypersurface of a ut sphere S +1 (1). If S + ɛ 1 (), the S =, whereɛ 1 ()= It s obvous that our pchg costat 17 Theore s larger tha thers.. Local forulae Let be a -desoal hypersurface a ut sphere S +1 (1). We choose a local orthooral frae feld {e 1,...,e +1 } S +1 (1), restrcted to, sothat e 1,...,e are taget to. Let ω 1,..., ω +1 deote the dual cofrae feld S +1 (1). The,, ω +1 =0. 17) It follows fro Carta s Lea that (.0) ω +1 = h j ω j, h j = h j. j The secod fudaetal for α ad the ea curvature of are defed by (.1) α =,j h j ω ω j e +1 ad H = h, respectvely. We recall that s by defto a al hypersurface f the ea curvature of s detcally zero. The coecto for ω j s characterzed by the Lcese or copyrght restrctos ay apply to redstrbuto; see

3 A CHARACTERIZATION OF THE CLIFFORD TORUS 81 structure equatos dω + j ω j ω j =0, ω j + ω j =0, (.) dω j + k ω k ω kj =Ω j, Ω j = 1 k,l R jklω k ω l, where Ω j (resp. R jkl ) deotes the curvature for (resp. the copoets of the curvature tesor) of. The Gauss equato s gve by (.3) R jkl =(δ k δ jl δ l δ jk )+(h k h jl h l h jk ). The covarat dervatve α of the secod fudaetal for α of wth copoets h jk s gve by h jk ω k = dh j + h jk ω k + h k ω jk. k k k The the exteror dervatve of (.0) together wth the structure equato yelds the followg Codazz equato: (.4) h jk = h kj = h jk. Fro the Codazz equato, we kow that h jk s syetrc the dces, j ad k. Slarly, we have the covarat dervatve α of α wth copoets h jkl as follows: h jkl ω l = dh jk + h ljk ω l + h lk ω jl + h jl ω kl, l l l l ad t s easy to get the followg Rcc forula: (.5) h jkl h jlk = h R jkl + h j R kl. Slarly, we also have (.6) h jkl h jkl = r h rjk R rl + r h rk R rjl + r h jr R rkl, where the h jkl s are the copoets of the covarat dervatve 3 α of α.we should reark that h jkl ad h jkl are syetrc the frst three dces, j ad k ad geerally ot syetrc the other oes. The copoets of the Rcc curvature ad the scalar curvature are gve by (.7) R j =( 1)δ j k h k h jk, (.8) R = ( 1),j h j. Now we copute certa local forulae. For ay fxed pot p, wecachoose a local orthooral frae feld e 1,...,e such that { 0 f j, (.9) h j = λ f = j. The followg forulas ca be obtaed by a drect coputato (cf. [1]). Let S :=,j h j = λ, Lcese or copyrght restrctos ay apply to redstrbuto; see

4 8 QING-ING CHENG AND SUSUU ISHIKAWA (.10) (.11) 1 S = h jk S(S ), 1 h jk = h jkl +(+3 S) h jk,l +3(B A) 3 S, where A = λ h jk ad B = λ λ j h jk. 3. Proofs of the theores At frst we gve two algebrac leas whch wll play a crucal role the proofs of our theores. Lea 1. Let a ( =1,,3,4)berealuberssatsfyg a =0ad a = a. The a4 7 1 a. Proof. We axze the fucto a4 wth the costrats a =0ad a = a. By eas of the ethod of the Lagrage ultpler, we solve the followg proble: f = a 4 + λ a + µ( a a), where λ ad µ are the Lagrage ultplers. The axu pot of a4 s a crtcal pot of f. Takg the dervatve of f wth respect to a,wehave f a =4a 3 +λ+µa =0. Hece, at ost three of the a s are dstct wth each other at a crtcal pot of f. We cosder the followg three cases. (1) Three of the a s are dstct wth each other. Wthout loss of geeralty, we deote the by a 1,a,a 3 ad assue a 1 = a 4 ;the Hece,.e., a 1 +a +a 3 =0, a 1+a +a 3 =a. a 4 =a4 1 +a4 +a4 3 =(a+a 1 ) = a +aa 1 1a a, a a. 14a 4 1 () Two of the a s are dstct wth each other. Wthout loss of geeralty, we deote the by a 1,a ad assue a 1 = a 4 ad a = a 3 or a 1 = a 3 = a 4 ;the a4 7 1 a. (3) If all of the a s are the sae, the a4 =0. Lcese or copyrght restrctos ay apply to redstrbuto; see

5 A CHARACTERIZATION OF THE CLIFFORD TORUS 83 Therefore, we coclude a a. Ths copletes the proof of Lea 1. Lea. Let a j ad b (, j =1,...,) be real ubers satsfyg b =0, b =b>0,,j b a j = 1 b( b) ad,j b ja j = 1 b( b). The a +3 a 3b( b) j ( +4). j Proof. We cosder F = a +3 j a j as a fucto of a j wth costrats,j b a j = 1 b( b) ad,j b ja j = 1 b( b). Let f := a +3 j a j + λ[,j b a j 1 b( b)] + µ[,j b j a j 1 b( b)], where λ ad µ are the Lagrage ultplers. It s obvous that the u pot of F s a crtcal pot of f. Takg the dervatve of f wth respect to a j,weget (3.1) f a =a + λb + µb =0, for, (3.) Hece ad Therefore, f aj =6a j + λb + µb j =0, for j. a f a = a + λ a b + µ a b =0 a j f aj =6 a j + λ a j b + µ a j b j =0. j j j j (3.3) [ a +3 j a j ]=λ1 b(b )+µ1 b(b ). Fro (3.1) ad (3.), we have (3.4) b a +(λ+µ) b =0, 6 b a j + λ b + µ b b j =0 j j j ad 6 b j a j + λ b b j + µ b j =0. j j j Fro (3.4) ad the two equaltes above, we get 4 b a +3b( b)+λb =0 Lcese or copyrght restrctos ay apply to redstrbuto; see

6 84 QING-ING CHENG AND SUSUU ISHIKAWA ad 4 b a +3b( b)+µb =0, Accordg to (3.3), we obta λ+µ= 6(b ) ( +4). 3b( b) f = ( +4). Thus we have fshed the proof of Lea. For ay fxed pot p, we ca choose a local frae feld e 1,...,e such that (3.5) h j = λ δ j. Defg f 3 = λ3 ad f 4 = λ4,thef 3 ad f 4 are fuctos defed globally o. Proposto 1. Let be a al hypersurface S +1 (1). The h jkl 3 (Sf 4 f3 S + S)+ 3S(S ) ( +4) holds.,l Proof. Fro the Rcc forula (.5) ad the Gauss equato (.3), we have (3.6) h jj h jj = h jj h jj = h R jj + h j R j We defe (3.7) = λ R jj + λ j R jj =(λ λ j )R jj =(λ λ j )(1 + λ λ j ). u jkl = 1 4 (h jkl + h ljk + h klj + h jkl ). Sce h jkl s syetrc the dces, j, k, fro forula (3.6), we obta (3.8) h jkl u jkl + 3 [Sf 4 f3 S + S].,l,l Sce h j =( S)h j ad h kl =0,wehave u jj λ = u jj λ j = 1 S( S).,j,j Fro λ =0ad λ =Sad defg a j := u jj ad b := λ,thea j ad b satsfy the codtos Lea. Fro the defto of u jkl, we kow that u jkl s syetrc the dces, j, k, l. Fro Lea, we fer (3.9) u jkl u +3 u 3S(S ) jj ( +4).,l j Lcese or copyrght restrctos ay apply to redstrbuto; see

7 A CHARACTERIZATION OF THE CLIFFORD TORUS 85 Hece, fro (3.8) ad (3.9), we obta h jkl 3 (Sf 4 f3 S + S)+ 3S(S ) ( +4).,l Ths copletes the proof of Proposto 1. Proposto. Let be a closed al hypersurface S +1 (1). The [(S 3 ) h jk +(S )f 4 3S(S ) + 9 ( +4) 8 S ]d 0 holds. Proof. The followg tegral forula (3.10) ca be foud []: (3.10) (A B)d = [Sf 4 S f3 1 4 S ]d. Fro the Rcc forula (.5), by a drect coputato, we obta 1 4 f 4 =( S)f 4 +A+B. Itegratg both sdes of the above equalty, we have (3.11) (S )f 4 d = (A + B)d. Forulas (3.10) ad (3.11) yeld (3.1) [(S 4)f 4 +3f3 +3S S ]d 0. Accordg to Stokes forula, we tegrate the forula (.11) ad obta (3.13) h jkld,l = +3 S) [ ( h jk 3(B A)+3 S ]d. Fro Proposto 1, (3.10) ad (3.13), we fer (3.14) {(S 3 ) h jk S + 3 [Sf 4 f3 S 3S( S) ] }d 0. ( +4) (3.1)+ (3.14) yelds [(S 3 ) h jk +(S )f 4 3S(S ) + 9 ( +4) 8 S ]d 0. Thus Proposto s vald. Proof of Theore 1. Accordg to (.10) ad Stokes Theore, we obta (3.15) h jk d = [S(S )]d Lcese or copyrght restrctos ay apply to redstrbuto; see

8 86 QING-ING CHENG AND SUSUU ISHIKAWA ad (3.16) 1 S = [S h jk +( S)S ]d. Fro forula (.7) ad the assupto Theore 1, we have Therefore, that s, R = 1 λ. λ, λ 4 λ, (3.17) f 4 S. Fro Proposto ad (3.17), we have {(S 3 ) h jk +( )S(S ) 3S(S ) ( +4) S }d 0. Fro (3.15), (3.16) ad the above equalty, we fer {( 5 4 S 7 ) h jk +[ 9 ) S 3(S ]S(S )}d 0. 4 ( +4) Sce 14( +4) S , we have {( 5 4 S 7 ) h jk +( )S(S )}d 0. Hece 5 4 (S ) h jkd =0. Sce S ad h jk are cotuous fuctos, we have S =. Thus, s soetrc to a Clfford torus S ( ) S ( )(1<< 1) fro the result due to Cher, do Caro ad Kobayash [3] or Lawso [4]. Ths copletes the proof of Theore 1. Proof of Theore. I the case = 3, because λ =0,wehavef 4 = λ4 = S. Fro Proposto, we have {(S 3 ) h jk + 3S(S S ) (S ) + 9 ( +4) 8 S }d 0. Lcese or copyrght restrctos ay apply to redstrbuto; see

9 A CHARACTERIZATION OF THE CLIFFORD TORUS 87 Fro (3.16) ad the above equalty, we have Sce we have {( 5 4 S 3 ) h jk S (S ) S , { 5 4 (S ) h jk }d 0. 3S(S ) }d 0. ( +4) Hece 5 4 (S ) h jkd =0. By akg use of the sae proof as Theore 1, we kow that Theore s true the case =3. I the case = 4, fro Lea 1, we have f S. By usg ths equalty, we obta, fro Proposto, {(S 3 ) h jk S (S ) 3S(S ) ( +4) S }d 0. By the sae proof as the case = 3, we kow that Theore s also vald the case =4. I the case = 5, fro Proposto 1, (3.10) ad (3.13), we have {(S 3 ) h jk + 3 3S(S ) (3.18) (A B) + 9 ( +4) 8 S }d 0. Sce 3(A B) = (λ + λ j + λ k λ λ j λ j λ k λ k λ )h jk = j k [(λ + λ j + λ k) (λ + λ j + λ k ) ]h jk +3 (λ k 4λ λ k )h k 3 k λ h S h jk, by akg use of ths equalty ad (3.18), a slar proof as the case =3 yelds that Theore s also vald ths case. We have fshed the proof of Theore. Ackowledgeet The frst aed author wshes to thak Professor K. Shohaa for hs costat ecourageet ad help. Lcese or copyrght restrctos ay apply to redstrbuto; see

10 88 QING-ING CHENG AND SUSUU ISHIKAWA Refereces 1. Cheg, Q.., The classfcato of coplete hypersurfaces wth costat ea curvature of space for of deso 4, e. Fac. Sc. KyushuUv. 47 (1993), R 94h:53067; Errata CP 95:01. Cheg, Q.., The rgdty of Clfford torus S 1 1 ( ) S 1 ( ( 1) ), Coet. ath. Helvetc 71 (1996), R 97a: Cher, S.S., do Caro,. ad Kobayash, S., al subafolds of a sphere wth secod fudaetal for of costat legth, Fuctoal aalyss ad related felds, Sprger, New York, 1970, pp R 4: Lawso, H.B., Local rgdty theores for al hypersurfaces, A. of ath. 89 (1969), R 38: Peg, C.K. ad Terg, C.L., The scalar curvature of al hypersurfaces spheres, ath. A. 66 (1983), R 85c: Yag, H.C. ad Cheg, Q.., A estate of the pchg costat of al hypersurfaces wth costat scalar curvature the ut spheres, auscrpta ath. 84 (1994), R 95c: Yag, H.C. ad Cheg, Q.., Cher s cojecture o al hypersurfaces, ath.z.7 (1998), CP 98:11 Departet of atheatcs, Faculty of Scece, Josa Uversty, Sakado, Sataa , Japa E-al address: cheg@ath.josa.ac.jp Departet of atheatcs, Saga Uversty, Saga , Japa Lcese or copyrght restrctos ay apply to redstrbuto; see