Hypersurfaces wth Costat Scalar Curvature a Hyperbolc Space Form Lu Xm ad Su Wehog Abstract Let M be a complete hypersurface wth costat ormalzed scalar curvature R a hyperbolc space form H +1. We prove that f R = R + 1 0 ad the orm square h 2 of the secod fudametal form of M satsfes R sup h 2 ( 2)( R 2) [( 1) R 2 4( 1) R + ], the ether sup h 2 = R ad M s a totally umblcal hypersurface; or sup h 2 = ( 2)( R 2) [( 1) R 2 4( 1) R + ], ad M s sometrc to S 1 (r) H 1 ( 1/(r 2 + 1)), for some r > 0. Mathematcs Subject Classfcato: 53C42, 53A10 Key words: hypersurface, hyperbolc space form, scalar curvature 1 Itroducto Let R +1 (c) be a ( + 1)-dmesoal Remaa mafold wth costat sectoal curvature c. We also call t a space form. Whe c > 0, R +1 (c) = S +1 (c) (.e. (+1)- dmesoal sphere space); whe c = 0, R +1 (c) = R +1 (.e. ( + 1)-dmesoal Eucldea space); whe c < 0, R +1 (c) = H +1 (c) (.e. ( + 1)-dmesoal hyperbolc space). We smply deote H +1 ( 1) by H +1. Let M be a -dmesoal hypersurface R +1 (c), ad e 1,..., e a local orthoormal frame feld o M, ω 1,..., ω ts dual coframe feld. The the secod fudametal form of M s (1) h =,j h j ω ω j. Further, ear ay gve pot p M, we ca choose a local frame feld e 1,..., e so that at p, h j ω ω j = k ω ω j, the the Gauss equato wrtes,j (2) R jj = c + k k j, j. Balka Joural of Geometry ad Its Applcatos, Vol.7, No.1, 2002, pp. 121-132. c Balka Socety of Geometers, Geometry Balka Press 2002.
122 L. Xm ad Su Wehog (3) ( 1)(R c) = 2 H 2 h 2, where R s the ormalzed scalar curvature, H = 1 k the mea curvature ad h 2 = k 2 the orm square of the secod fudametal form of M. As t s well kow, there are may rgdty results for mmal hypersurfaces or hypersurfaces wth costat mea curvature H R +1 (c) (c 0) by use of J. Smos method, for example, see [1], [4], [5], [8], [12] etc., but less were obtaed for hypersurfaces mmersed to a hyperbolc space form. Walter [13] gave a classfcato for o-egatvely curved compact hypersurfaces a space form uder the assumpto that the rth mea curvature s costat. Morva-Wu [7], Wu [14] also proved some rgdty theorems for complete hypersurfaces M a hyperbolc space form H +1 (c) uder the assumpto that the mea curvature s costat ad the Rcc curvature s o-egatve. Moreover, they proved that M s a geodesc dstace sphere H +1 (c) provded that t s compact. O the other had, Cheg-Yau [3] troduced a ew self-adjot dfferetal operator to study the hypersurfaces wth costat scalar curvature. Later, L [6] obtaed terestg rgdty results for compact hypersurfaces wth costat scalar curvature space-forms usg the Cheg-Yau s self-adjot operator. I the preset paper, we use Cheg-Yau s self-adjot operator to study the complete hypersurfaces a hyperbolc space form wth costat scalar curvature, ad prove the followg theorem: Theorem. Let M be a -dmesoal ( 3) complete hypersurface wth costat ormalzed scalar curvature R H +1. If (1) R = R + 1 0, (2) the orm square h 2 of the secod fudametal form of M satsfes the ether R sup h 2 ( 2)( R 2) [( 1) R 2 4( 1) R + ], sup h 2 = R ad M s a totally umblcal hypersurface; or sup h 2 = ( 2)( R 2) [( 1) R 2 4( 1) R + ], ad M s sometrc to S 1 (r) H 1 ( 1/(r 2 + 1)), for some r > 0. 2 Prelmares Let M be a -dmesoal hypersurface H +1. We choose a local orthoormal frame e 1,..., e +1 H +1 such that at each pot of M, e 1,..., e spa the taget space of M ad form a orthoormal frame there. Let ω 1,..., ω +1 be ts dual coframe. I ths paper, we use the followg coveto o the rage of dces:
Hypersurfaces wth Costat Scalar Curvature 123 1 A, B, C,... + 1; 1, j, k,.... The the structure equatos of H +1 are gve by (4) dω A = B ω AB ω B, ω AB + ω BA = 0, (5) dω AB = C ω AC ω CB 1 K ABCD ω C ω D, 2 C,D (6) K ABCD = (δ AC δ BD δ AD δ BC ). Restrctg these forms to M, we have (7) ω +1 = 0. From Carta s lemma we ca wrte (8) ω +1 = j h j ω j, h j = h j. From these formulas, we obta the structure equatos of M : (9) dω = j ω j ω j, ω j + ω j = 0, (10) dω j = k ω k ω kj 1 R jkl ω k ω l, 2 k,l (11) R jkl = (δ k δ jl δ l δ jk ) + (h k h jl h l h jk ), where R jkl are the compoets of the curvature tesor of M ad (12) h =,j h j ω ω j s the secod fudametal form of M. We also have (13) R j = ( 1)δ j + Hh j k h k h kj, (14) ( 1)(R + 1) = 2 H 2 h 2, where R s the ormalzed scalar curvature, ad H the mea curvature. Defe the frst ad the secod covarat dervatves of h j, say h jk ad h jkl by (15) h jk ω k = dh j + h kj ω k + h k ω kj, k k k (16) h jkl ω l = dh jk + h mjk ω m + h mk ω mj + h jm ω mk. m m m l The we have the Codazz equato
124 L. Xm ad Su Wehog (17) h jk = h kj, ad the Rcc detty (18) h jkl h jlk = m h mj R mkl + m h m R mjkl. For a C 2 -fucto f defed o M, we defed ts gradet ad Hessa (f j ) by the followg formulas df = (19) f ω, f j ω j = df + f j ω j. j j The Laplaca of f s defed by f = f. Let φ = φ j ω ω j be a symmetrc tesor defed o M, where j (20) φ j = Hδ j h j. Followg Cheg-Yau [3], we troduce a opertator assocated to φ actg o ay C 2 -fucto f by (21) f =,j φ j f j =,j (Hδ j h j )f j. Sce φ j s dvergece-free, t follows [3] that the operator s self-adjot relatve to the L 2 er product of M,.e. (22) f g = g f. M M We ca choose a local frame feld e 1,....e at ay pot p M, such that h j = k δ j at p, by use of (21) ad (14), we have (H) = H (H) k (H) = (23) = 1 2 (H)2 (H) 2 k (H) = = 1 2 ( 1) R + 1 2 h 2 2 H 2 k (H). O the other had, through a stadard calculato by use of (17) ad (18), we get (24) 1 2 h 2 = h 2 jk +,j,k Puttg (24) to (23), we have (25) k (H) + 1 R jj (k k j ) 2. 2 (H) = 1 2 ( 1) R + h 2 2 H 2 + 1 R jj (k k j ) 2. 2,j,j
Hypersurfaces wth Costat Scalar Curvature 125 From (11), we have R jj = 1 + k k j, j, ad by puttg ths to (25), we obta (26) (H) = 1 2 ( 1) R + h 2 2 H 2 h 2 + 2 H 2 h 4 + H k 3. Let µ = k H ad Z 2 = µ 2, we have (27) (28) From (26)-(28), we get µ = 0, Z 2 = h 2 H 2, k 3 = µ 3 + 3H Z 2 + H 3, (29) (H) = 1 2 ( 1) R+ h 2 2 H 2 + Z 2 ( +H 2 Z 2 )+H µ 3. We eed the followg algebrac lemma due to M. Okumura [9] (see also [1]). Lemma 2.1. Let µ, = 1,...,, be real umbers such that µ = 0 ad µ 2 = β 2, where β = costat 0. The (30) 2 ( 1) β 3 µ 3 2 ( 1) β 3, ad the equalty holds (30) f ad oly f at least ( 1) of the µ are equal. By use of Lemma 2.1, we have (H) 1 2 ( 1) R + h 2 2 H 2 + ( + ( h 2 H 2 ) + 2H 2 h 2 ( 2) H ) (31) h 2 H 2. ( 1) 3 Umblcal hypersurface a hyperbolc space form I ths secto, we cosder some specal hypersurfaces a hyperbolc space form whch we wll eed the followg dscusso. Frst we wat to gve a descrpto of the real hyperbolc space-form H +1 (c) of costat curvature c (< 0). For ay two vectors x ad y R +2, we set +1 g(x, y) = x y x +2 y +2. =1 (R +2, g) s the so-called Mkowsk space-tme. Deote ρ = 1/c. We defe
126 L. Xm ad Su Wehog H +1 (c) = {x R +2 x +2 > 0, g(x, x) = ρ 2 }. The H +1 (c) s a coected smply-coected hypersurface of R +2. It s ot hard to check that the restrcto of g to the taget space of H +1 (c) yelds a complete Remaa metrc of costat curvature c. Here we obta a model of a real hyperbolc space form. We are terested those complete hypersurfaces wth at most two dstct costat prcpal curvatures H +1 (c). Ths kd of hypersurfaces was descrbed by Lawso [5] ad completely classfed by Rya [11]. Lemma 3.1 [11]. Let M be a complete hypersurface H +1 (c). Suppose that, uder a sutable choce of a local orthoormal taget frame feld of T M, the shape operator over T M s expressed as a matrx A. If M has at most two dstct costat prcpal curvatures, the t s cogruet to oe of the followg: (1) M 1 = {x H +1 (c) x 1 = 0}. I ths case, A = 0, ad M 1 s totally geodesc. Hece M 1 s sometrc to H (c); (2) M 2 = {x H +1 (c) x 1 = r > 0}. I ths case, A = 1/ρ 2 1/ρ2 + 1/r 2 I, where I deotes the detty matrx of degree, ad M 2 s sometrc to H ( 1/(r 2 + ρ 2 )); (3) M 3 = {x H +1 (c) x +2 = x +1 + ρ}. I ths case, A = 1 ρ I, ad M 3 s sometrc to a Eucldea space E ; +1 (4) M 4 = {x H +1 (c) x 2 = r 2 > 0}. I ths case, A = 1/r 2 + 1/ρ 2 I, =1 ad M 4 s sometrc to a roud sphere S (r) of radus r; k+1 +1 (5) M 5 = {x H +1 (c) x 2 = r 2 > 0, x 2 j x 2 +2 = ρ 2 r 2 }. =1 j=k+2 I ths case, A = λi k νi k, where λ = 1/ρ 2 + 1/r 2, ad ν = s sometrc to S k (r) H k ( 1/(r 2 + ρ 2 )). 1/ρ 2 1/r2 + 1ρ 2, M 5 Remark 3.1. M 1,..., M 5 are ofte called the stadard examples of complete hypersurfaces H +1 (c) wth at most two dstct costat prcpal curvatures. It s obvous that M 1,..., M 4 are totally umblcal. I the sece of Che [2], they are called the hyperspheres of H +1 (c). M 3 s called the horosphere ad M 4 the geodesc dstace sphere of H +1 (c). Remark 3.2. Rya [11] stated that the shape operator of M 2 s A = 1/r 2 1/ρ 2 I, ad M 2 s sometrc to H ( 1/r 2 ), where r ρ. Ths s correct ad we have corrected t here. 4 The proof of Theorem The followg lemma essetally due to Cheg-Yau [3].
Hypersurfaces wth Costat Scalar Curvature 127 Lemma 4.1. Let M be a -dmesoal hypersurface H +1.Suppose that the ormalzed scalar curvature R = costat ad R 1. The h 2 2 H 2. Proof. From (14), 2 H 2 h 2 j = ( 1)(R + 1).,j Takg the covarat dervatve of the above expresso, ad usg the fact R = costat, we get 2 HH k = h j h jk.,j By Cauchy-Schwarz equalty, we have 4 H 2 (H k ) 2 = ( k k,j h j h jk ) 2 (,j h 2 j),j,k h 2 jk, that s 4 H 2 H 2 h 2 h 2. O the other had, from R + 1 0, we have 2 H 2 h 2 0. Thus H 2 h 2 2 H 2 H 2 ad Lemma 4.1 follows. From the assumpto of the Theorem that R s costat ad R = R + 1 0 ad Lemma 4.1 we have ( (H) ( h 2 H 2 ) + 2H 2 h 2 ( 2) H ) (32) h 2 H 2. ( 1) By Gauss equato (14) we kow that (33) From (32) ad (33) we have Z 2 = h 2 H 2 = 1 ( h 2 R). (34) where (H) 1 ( h 2 R)φ H ( h ), φ H ( h ) = + 2H 2 h 2 ( 2) ( 1) H h 2 H 2. By (33) we ca wrte φ H ( h ) as φ R( h ) = +2( 1) R 2 h 2 2 (35) (( 1) R + h 2 )( h 2 R). Therefore (34) becomes (36) (H) 1 ( h 2 R)φ R( h ),
128 L. Xm ad Su Wehog (37) It s a drect check that our assumpto s equvalet to (38) sup h 2 ( +2( 1) R 2 sup h 2 ) 2 ( 2)( R 2) [( 1) R 2 4( 1) R + ] But t s clear from (37) that (38) s equvalet to (39) ( 2)2 2 (( 1) R+sup h 2 )(sup h 2 R). +2( 1) R 2 sup h 2 2 (( 1) R + sup h 2 )(sup h 2 R). So uder the hypothess that sup h 2 ( 2)( R 2) [( 1) R 2 4( 1) R + ], we have (40) O the other had, φ R( sup h 2 ) 0. (H) =,j (Hδ j h j )(H) j = (H h )(H) = (41) = H(H) k (H) ( H max C) (H), where H max s the maxmum of the mea curvature H ad C = m k s the mmum of the prcpal curvatures of M. Now we eed the followg maxmum prcple at fty for complete mafolds due to Omor [10] ad Yau [15]: Lemma 4.2. Let M be a -dmesoal complete Remaa mafold whose Rcc curvature s bouded from below ad f : M R a smooth fucto bouded from below. The for each ε > 0 there exsts a pot p ε M such that () f (p ε ) < ε, () f(p ε ) > ε, () f f f(p ε ) f f + ε. From the hypothess of the Theorem ad Gauss equato, we kow that the Rcc curvature s bouded below. So we may apply Lemma 4.2 to the followg smooth fucto f o M defed by f = 1 1 + (H) 2. It s mmedate to check that
Hypersurfaces wth Costat Scalar Curvature 129 (42) f 2 = 1 (H) 2 2 4 (1 + (H) 2 ) 3 ad that f = 1 (H) 2 2 (1 + (H) 2 ) + 3 (H) 2 2 (43) 3/2 4 (1 + (H) 2 ). 5/2 By Lemma 4.2 we ca fd a sequece of pots p k, k N M, such that (44) lm f(p k) = f f, f(p k ) > 1 k k, f 2 (p k ) < 1 k 2. Usg (44) the equatos (42) ad (43) ad the fact that (45) lm k (H)(p k) = sup p M (H)(p), we get 1 k 1 (H) 2 (46) 2 (1 + (H) 2 ) (p k) + 3 3/2 k 2 (1 + (H)2 (p k )) 1/2. Hece we obta (47) (H) 2 (1 + (H) 2 ) 2 (p k) < 2 k ( 1 1 + (H)2 (p k ) + 3 k ). O the other had, by (36) ad (41), we have (48) 1 ( h 2 R)φ R( h ) (H) ( H max C) (H). At pots p k of the sequece gve (44), ths becomes (49) 1 ( h 2 (p k ) R)φ R( h (p k )) (H(p k )) ( H max C) (H)(p k ). Let k ad use (47) we have that the rght had sde of (49) goes to zero, so we have ether 1 (sup h 2 R) = 0,.e. sup h 2 = R or φ R( sup h 2 ) = 0. If sup h 2 = R, by (33) Z 2 = 1 ( h 2 R) we have sup Z 2 = 1 (sup h 2 R) = 0, the Z 2 = 0 ad M s totally umblcal. If φ R( sup h 2 ) = 0, t s easy to prove that sup H 2 = 1 [ 2 ( 1) 2 R ] + 2 2 2( 1) + 2 R, + 2 the equaltes hold (30) ad Lemma 4.1, we follow that k = costat for all ad ( 1) of the k are equal. After reumberato f ecessary, we ca assume k 1 = k 2 = = k 1 k 1 k.
130 L. Xm ad Su Wehog Therefore, from Lemma 3.1, we kow that M s a hypersurface H +1 wth two dstct prcpal curvatures, ad M s sometrc to S 1 (r) H 1 ( 1/(r 2 + 1)), for some r > 0. Ths completes the proof of Theorem. Whe M s compact, we ca prove Corollary 1. Let M be a -dmesoal ( 3) compact hypersurface wth costat ormalzed scalar curvature R H +1. If (1) R = R + 1 0, (2) the orm square h 2 of the secod fudametal form of M satsfes (50) R h 2 ( 2)( R 2) [( 1) R 2 4( 1) R + ], the M s a totally umblcal hypersurface. Proof. From (36) we have (51) (H) 1 ( h 2 R)[ + 2( 1) R 2 h 2 2 (( 1) R + h 2 )( h 2 R)], It s a drect check that our assumpto codto (50) s equvalet to ( + 2( 1) R 2 ) 2 (52) h 2 But t s clear from (50) that (52) s equvalet to (53) ( 2)2 2 (( 1) R + h 2 )( h 2 R). + 2( 1) R 2 h 2 2 (( 1) R + h 2 )( h 2 R), therefore the rght had sde of (51) s o-egatve. Because M s compact ad the operator s self-adjot, we have M (H)dv = 0. Thus ether (54) h 2 = R, that s, h 2 = H 2, M s a totally umblcal hypersurface; or (55) h 2 = ( 2)( R 2) [( 1) R 2 4( 1) R + ]. I the latter case, equaltes hold (30) ad Lemma 4.1, ad t follows that M has at most two dstct costat prcpal curvatures. We coclude that M s totally umblcal from the compactess of M. Ths completes the proof of Corollary 1. Corollary 2. Let M be a -dmesoal compact hypersurface wth costat ormalzed scalar curvature R ad R + 1 0 H +1. If M has o-egatve sectoal curvature, the M s a totally umblcal hypersurface. Proof. Because M s compact ad the operator s self-adjot, form (25), we have
Hypersurfaces wth Costat Scalar Curvature 131 (56) M h 2 2 H 2 + 1 R jj (k k j ) 2 = 0. 2 If M has costat ormalzed scalar curvature R ad R 1, from Lemma 4.1, we have h 2 2 H 2. So f M has o-egatve sectoal curvature, form (56) we have h 2 = 2 H 2 ad R jj = 0, whe k = k j o M. Sce R jj = 1 + k k j, the ether M s totally umblcal, or M has two dfferet prcpal curvatures, the latter case, M s stll totally umblcal from the compactess of M. Ths completes the proof of Corollary 2.,j Refereces [1] H. Alecar ad M. P. do Carmo, Hypersurfaces wth costat mea curvature hyperbolc space forms, Proc. Amer. Math. Soc. 120 (1994), 1223-1229. [2] B.Y. Che, Totally mea curvature ad submafolds of fte type, World Scetfc, Sgapore, 1984. [3] S.Y. Cheg ad S.T. Yau, Hypersurfaces wth costat scalar curvature, Math. A., 225 (1977), 195-204. [4] Z. H. Hou, Hypersurfaces sphere wth costat mea curvature, Proc. Amer. Math. Soc., 125 (1997), 1193-1196. [5] H.B. Lawso, Jr., Local rgdty theorems for mmal hypersurfaces, A. of Math. (2), 89 (1969), 187-197. [6] H. L, Hypersurfaces wth costat scalar curvature space forms, Math. A. 305 (1996), 665-672. [7] J.M. Morva ad B.Q. Wu, Hypersurfaces wth costat mea curvature hyperbolc space form, Deom. Dedcata, 59 (1996), 197-222. [8] K. Nomzu ad B. Smyth, A formula for Smo s type ad hypersurfaces, J. Dff. Geom., 3 (1969), 367-377. [9] M. Okumuru, Hypersurfaces ad a pchg problem o the secod fudametal tesor, Amer. J. Math., 96 (1974), 207-213. [10] H. Omor, Isometrc mmersos of Remaa mafolds, J. Math. Soc. Japa, 19 (1967), 205-214. [11] P.J. Rya, Hypersurfaces wth parallel Rcc tesor, Osaka J. Math. 8 (1971), 251-259. [12] J. Smos, Mmal varetes Remaa mafolds, A. of Math. (2) 88 (1968), 62-105. [13] R. Walter, Compact hypersurfaces wth a costat hgher mea curvature fucto, Math. A., 270 (1985), 125-145.
132 L. Xm ad Su Wehog [14] B.Q. Wu, Hypersurface wth costat mea curvature H +1, The Math. Hertage of C. F. Gauss, World Scetfc, Sgapore, 1991, 862-871. [15] S.T. Yau, Harmoc fuctos o complete Remaa mafolds, Comm. Pure ad Appl. Math., 28(1975), 201-228. Departmet of Mathematcal Sceces Rutgers Uversty, Camde, New Jersey 08102, USA emal: xmlu@camde.rutgers.edu Departmet of Appled Mathematcs Bejg Isttute of Techology Bejg 100081, Cha