COMPLETE SPACELIKE HYPERSURFACES IN THE DE SITTER SPACE

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1 Chao, X. Osaka J. Math. 50 (203), COMPLETE SPACELIKE HYPERSURFACES IN THE DE SITTER SPACE XIAOLI CHAO (Received August 8, 20, revised December 7, 20) Abstract In this paper, by modifying Cheng Yau s technique to complete spacelike hypersurfaces in the de Sitter (n )-space S n (), we prove a rigidity under the hypothesis of the mean curvature and the normalized scalar curvature being linearly related. As a corollary, we have the Theorem. of [3].. Introduction Let L n2 be the (n 2)-dimensional Lorentz Minkowski space, that is, the real vector space R n2 endowed with the Lorentzian metric (.) Ú, Û Ú 0 Û 0 n for Ú, Û ¾ R n2. Then, the (n )-dimensional de Sitter space S n () can be defined as the following hyperquadric of L n2 i Ú i Û i (.2) S n () {x ¾ L n2 Ï x, x }. A smooth immersion ³Ï M n S n () L n2 of an n dimensional connected manifold M n is said to be a spacelike hypersurface if the induced metric via ³ is a Riemannian metric on M n. As is uaual, the spacelike hypersurface is said to be complete if the Riemannian induced metric is a complete metric on M n. By endowing M n with the induced metric we can suppose M n to be Riemannian and ³ to be an isometric spacelike immersion. The interest in the study of spacelike hypersurfaces immersed in the de Sitter space is motivated by their nice Bernstein-type properties. It was proved by E. Calabi [2] (for n 4) and by S.Y. Cheng and S.T. Yau [7] (for all n) that a complete maximal spacelike hypersurface in L n2 is totally geodesic. In [3], S. Nishikawa obtained similar results for others Lorentzian manifolds. In particular, he proved that a complete maximal spacelike hypersurface in S n () is totally geodesic Mathematics Subject Classification. 53C40, 53C42. Research supported by grant No of NSFC.

2 76 X. CHAO Goddard [9] conjectured that complete spacelike hypersurface with constant mean curvature in the de Sitter S n () should be umbilical. Although the conjecture turned out to be false in its original statement, it motivated a great deal of work of several authors trying to find a positive answer to the conjecture under appropriate additional hypotheses. For instance, in 987 Akutagawa [] proved the Goddard conjecture when H 2 if n 2 (see also Ramanathan [6]) and H 2 4(n )n 2 if n 2. He and Ramanathan also showed that when n 2, for any constant H 2 c 2, there exists a non-umbilical surface of mean curvature H in the de Sitter space S 3 (c) of constant curvature c 0. One year later, S. Montiel [] (and Akutagawa [], Ramanathan [6] when n 2) solved Goddard s problem in the compact case in S n () without restriction over the range of H. He also gave examples of non-umbilical complete spacelike hypersurfaces in S n () with constant H satisying H 2 4(n )n 2 if n 2, including the so-called hyperbolic cylinders. In [2], Montiel proved that the only complete spacelike hypersurface in S n () with constant H 2 Ô n n with more than one topological end is a hyperbolic cylinder. At the same time, the complete hypersurfaces in the de Sitter space have been characterized by Cheng [5] under the hypothesis of the mean curvature and the scalar curvature being linearly related. On the other hand, for the study of spacelike hypersurfaces with constant scalar curvature in de Sitter spaces, Y. Zheng [8] proved that a compact spacelike hypersurface in S n () with constant normalized scalar curvature r, r and non-negative sectional curvatures is totally umbilical. Later, Q.M. Cheng and S. Ishikawa [6] showed that Zheng s result in [8] is also true without additional assumptions on the sectional curvatures of the hypersurface. In [0], H. Li proposed the following problem: Let M n be a complete spacelike hypersurface in S n (), n 3, with constant normalized scalar curvature r satisfying (n 2)n r. Is M n totally umbilical? A. Caminha [4] answered that question affirmatively under the additional condition that the supremum of H is attained on M n. Recently, Camargo Chaves Sousa [3] showed that Li s question is also true if the mean curvature is bounded. In this paper, by modifying Cheng Yau s technique to complete spacelike hypersurfaces in S n (), we prove a rigidity theorem under the hypothesis of the mean curvature and the normalized scalar curvature being linearly related. More precisely, we have Theorem.. Let M n be a complete spacelike hypersurface of S n () with bounded mean curvature. If r ah b, a, b ¾ Ê, a 0, b (n 2)n, (n )a 2 4n( b) 0, then M n is totally umbilical. If we choose a 0 and (n 2)n b in Theorem., we obtain the Theorem. of [3]: Corollary.2. Let M n be a complete spacelike hypersurface of S n () with constant normalized scalar curvature r satisfying (n 2)n r. If M n has bounded

3 SPACELIKE HYPERSURFACES IN THE DE SITTER SPACE 77 mean curvature, then M n is totally umbilical. 2. Preliminaries Let M be an n-dimensional complete spacelike hypersurface of de Sitter space S n (). For any p ¾ M, we choose a local orthonormal frame e,, e n, e n in S n () around p such that e,, e n are tangent to M. Take the corresponding dual coframe,, n, n, We use the following standard convention for indices: A, B, C, D, n, i, j, k, l, n. Let i, n, then the structure equations of S n () are given by (2.) d A B B AB B, AB B A 0, (2.2) (2.3) d AB C C AC C B 2 n C, D R ABC D A B (Æ AC Æ B D Æ AD Æ BC ). R ABC D C D, Restricting those forms to M, we get the structure equations of M (2.4) (2.5) The Gauss equations are d i n j d i j n k i j j, i j ji 0, ik k j 2 n k,l R i jkl k l. (2.6) (2.7) R i jkl (Æ ik Æ jl Æ il Æ jk ) (h ik h jl h il h jk ), n(n )r n(n ) n 2 H 2 B 2, where r ((n(n ))) È i, j R i ji j is the normalized scalar curvature of M and the norm square of the second fundamental form is (2.8) B 2 i, j h 2 i j. The Codazzi equations are (2.9) h i jk h ik j h jik,

4 78 X. CHAO where the covariant derivative of h i j (2.0) k h i jk k dh i j k is defined by h k j ki k h ik k j. Similarly, the components h i jkl of the second derivative Ö 2 h are given by (2.) l h i jkl l dh i jk l h l jk li h ilk l j l l h i jl lk. By exterior differentiation of (2.0), we can get the following Ricci formula (2.2) h i jkl h i jlk m h im R mjkl m h jm R mikl. The Laplacian h i j of h i j is defined by h i j È k h i jkk, from the Codazzi equation and Ricci formula, we have (2.3) h i j k h kki j m,k h km R mi jk m,k h im R mk jk. Set i j h i j H Æ i j. It is easy to check that is traceless and 2 È i, j 2 i j B 2 nh 2. Following Cheng Yau [8], for each a 0, we introduce a modified operator acting on any C 2 -function f by (2.4) ( f ) i, j where f i j is given by the following Lemma 2.. j nh n 2 a Æ i j h i j f i j, f i j j d f i f j i j. Let M n be a complete spacelike hypersurface of S n () with r ah b, a, b ¾ Ê and (n )a 2 4n 4nb 0. Then we have (2.5) Ö B 2 n 2 Ö H 2. Proof. From Gauss equation, we have B 2 n 2 H 2 n(n )(r ) n 2 H 2 n(n )(ah b ). Taking the covariant derivative of the above equation, we have 2 i, j h i j h i jk 2n 2 H H k n(n )ah k.

5 SPACELIKE HYPERSURFACES IN THE DE SITTER SPACE 79 Therefore, 4B 2 Ö B 2 4 k ¼ i, j h i j h i jk ½ 2 [2n 2 H n(n )a] 2 Ö H 2. Since we know [2n 2 H n(n )a] 2 4n 2 B 2 4n 4 H 2 n 2 (n ) 2 a 2 4n 3 (n )ah it follows that Lemma 2.2. ah b, a, b ¾ Ê. Then we have 4n 2 (n 2 H 2 n(n )(ah b )) n 2 (n ) 2 a 2 4n 3 (n )(b ) n 2 (n )[(n )a 2 4n 4nb] 0, Ö B 2 n 2 Ö H 2. Let M n be a complete spacelike hypersurface of S n () with r (2.6) (nh) i, j,k h 2 i jk n 2 Ö H n(n 2) Ô n(n ) Hn( H 2 ). Proof. First, from (2.6) and (2.2), we have 2 B2 2 i, j h 2 i j i, j h i j h i j i, j,k h 2 i jk i, j,k h 2 i jk n i, j h i j H i j n(b 2 nh 2 ) nh f 3 B 4, where f 3 È i, j,k h i jh jk h ki. On the other side, from Gauss equation and r ah b, we have (2.7) B 2 (n 2 H 2 n(n )(r )) (n 2 H 2 n(n )(ah b )) (n 2 H 2 (n )anh) (nh 2 (n )a)2.

6 720 X. CHAO Then from (2.7) and Okumura s inequality [4], we get (nh) i, j ((nh 2 (n )a)æ i j h i j )(nh) i j (nh 2 (n )a)(nh) i, j (nh 2 (n )a)(nh 2 (n )a) i, j 2 nh 2 (n )a 2 Ö(nH 2 (n )a) nh (n )a n Ö H 2 2 i, j 2 i, j 2 B2 n 2 Ö H 2 i, j i, j,k i, j,k h 2 i jk n 2 Ö H 2 n(b 2 nh 2 ) nh f 3 B 4 hi 2 jk n Ö H n(n 2) Ô n(n ) Hn( H 2 ) Proposition 2.3. Let M n be a complete spacelike hypersurface of S n () with bounded mean curvature. If r ah b, a, b ¾ Ê, a 0, (n )a 2 4n 4nb 0, then there is sequence of points {p k } ¾ M n such that lim nh(p k) n sup HÁ k½ lim ÖnH(p k) 0Á k½ lim sup( (nh)(p k )) 0. k½ Proof. Choose a local orthonormal frame field e,, e n at p ¾ M n such that h i j i Æ i j. Thus (nh) i nh 2 (n )a i (nh) ii.. If H 0 the proposition is obvious. Let us suppose that H is not identically zero. By changing the orientation of M n if necessary, we may assume sup H 0. From 2 i B 2 n 2 H 2 n(n )(ah b ) (nh) 2 (n )a(nh)n(n )(b ) nh 2 (n )a 2 4 (n )((n )a2 4n 4nb) nh 2 (n )a 2,

7 SPACELIKE HYPERSURFACES IN THE DE SITTER SPACE 72 we have (2.8) i nh 2 (n )a. Then, for i, j with i j, (2.9) R i ji j i j nh 2 (n )a 2. Because H is bounded, it follows from (2.9) that the sectional curvatures are bounded from below. Therefore we may apply generalized maximun principle [5] [7] to nh, obtaining a sequence of points {p k } ¾ M n such that (2.20) lim k½ nh(p k) n sup HÁ lim ÖnH(p k) 0Á k½ lim sup((nh) ii (p k )) 0. k½ Since H is bounded, taking subsequences if necessary, we can arrive to a sequence {p k } ¾ M n which satisfies (2.20) and such that H(p k ) 0. Thus from (2.8) we get (2.2) 0 nh(p k ) 2 (n )a i(p k ) nh(p k ) 2 (n )a i(p k ) nh(p k ) 2 (n )a i(p k ) 2nH(p k )(n )a. Using once more the fact that H is bounded, from (2.2) we infer that nh(p k ) (2)(n )a i (p k ) is non-negative and bounded. By applying (nh) at p k, taking the limit and using (2.20) and (2.2) we have lim sup( (nh)(p k )) k½ i 3. Proof of results 0. lim sup k½ nh 2 (n )a i (p k )(nh) ii (p k ) Proof of Theorem.. If M n is maximal, i.e., if H 0, due to Nishikawa s result [3], we know that M n is totally geodesic. Let us suppose that H is not identically zero. In this case, by Proposition 2.3, it is possible to obtain a sequence of points {p k } ¾ M n such that (3.) lim sup( (nh)(p k )) 0, k½ lim H(p k) sup H 0. k½ Moreover, using the Gauss equation, we have that (3.2) 2 B 2 nh 2 n(n )(H 2 ah b ).

8 722 X. CHAO In view of lim k½ H(p k ) sup H and a 0, (3.2) implies that lim k½ 2 (p k ) sup 2. Now we consider the following polynomial given by 2 n(n 2) (3.3) P sup H (x) x Ô n(n ) sup H x n( sup H 2 ). If sup H 2 4(n )n 2, then the discriminant of P sup H (x) is negative. Hence, P sup H (sup) 0. If sup H 2 4(n )n 2, let the biggest root of P sup H (x) 0, which is positive. It s easy to check that sup provided a 0 and b (n 2)n. In fact, (3.4) (sup) 2 sup 2 n(n )(sup H 2 a sup H b ) n(n )(sup H 2 b ), it is straightforward to verify that (3.5) sup 2 2 n 2 2(n ) n 2 sup H 2 n sup H Ô n 2 sup H 2 4(n )c where c(2n(n )(n 2))((n )b (n 2)). It can be easily seen that sup if and only if n 2 sup H 2 n sup H Ô n 2 sup H 2 4(n )c 0. So, if b (n 2)n, the last inequality is true. In fact, it s true if and only if (3.6) (n 2 sup H 2 c) 2 n 2 sup H 2 (n 2 sup H 2 4(n )), i.e. (3.7) n 2 sup H 2 (2c4(n ))c 2 0. If b (n 2)n, then 2c4(n ) 0. Hence n 2 sup H 2 (2c4(n ))c Then we deduce that P sup H (sup) 0. Using Lemma 2. and evaluating (2.6) at the points p k the limit and using (3.), we obtain that of the sequence, taking 0 lim sup( (nh)(p k )) sup 2 P sup H (sup) 0, k½ and so sup 2 P sup H (sup) 0. Therefore, since P sup H (sup) 0, we conclude that sup 2 0 which shows that M n is totally umbilical.

9 SPACELIKE HYPERSURFACES IN THE DE SITTER SPACE 723 ACKNOWLEDGEMENT. This work was done while the author was visiting University of Notre Dame. The author is grateful for the hospitality of the mathematical department of University of Notre Dame. We would like to express our gratitude to the referee for valuable suggestions. References [] K. Akutagawa: On spacelike hypersurfaces with constant mean curvature in the de Sitter space, Math. Z. 96 (987), 3 9. [2] E. Calabi: Examples of Bernstein problems for some nonlinear equations, Math. Proc. Cambridge Phil. Soc. 82 (977), [3] F.E.C. Camargo, R.M.B. Chaves and L.A.M. Sousa, Jr.: Rigidity theorems for complete spacelike hypersurfaces with constant scalar curvature in de Sitter space, Differential Geom. Appl. 26 (2008), [4] A. Caminha: A rigidity theorem for complete CMC hypersurfaces in Lorentz manifolds, Differential Geom. Appl. 24 (2006), [5] Q.M. Cheng: Complete space-like hypersurfaces of a de Sitter space with r k H, Mem. Fac. Sci. Kyushu Univ. Ser. A 44 (990), [6] Q.-M. Cheng and S. Ishikawa: Spacelike hypersurfaces with constant scalar curvature, Manuscripta Math. 95 (998), [7] S.Y. Cheng and S.T. Yau: Maximal space-like hypersurfaces in the Lorentz Minkowski spaces, Ann. of Math. (2) 04 (976), [8] S.Y. Cheng and S.T. Yau: Hypersurfaces with constant scalar curvature, Math. Ann. 225 (977), [9] A.J. Goddard: Some remarks on the existence of spacelike hypersurfaces of constant mean curvature, Math. Proc. Cambridge Philos. Soc. 82 (977), [0] H. Li: Global rigidity theorems of hypersurfaces, Ark. Mat. 35 (997), [] S. Montiel: An integral inequality for compact spacelike hypersurfaces in de Sitter space and applications to the case of constant mean curvature, Indiana Univ. Math. J. 37 (988), [2] S. Montiel: A characterization of hyperbolic cylinders in the de Sitter space, Tohoku Math. J. (2) 48 (996), [3] S. Nishikawa: On maximal spacelike hypersurfaces in a Lorentzian manifold, Nagoya Math. J. 95 (984), [4] M. Okumura: Hypersurfaces and a pinching problem on the second fundamental tensor, J. Math. Soc. Japan 9 (967), [5] H. Omori: Isometric immersions of Riemannian manifolds, J. Math. Soc. Japan 9 (967), [6] J. Ramanathan: Complete spacelike hypersurfaces of constant mean curvature in de Sitter space, Indiana Univ. Math. J. 36 (987), [7] S.T. Yau: Harmonic functions on complete Riemannian manifolds, Comm. Pure Appl. Math. 28 (975), [8] Y. Zheng: Space-like hypersurfaces with constant scalar curvature in the de Sitter spaces, Differential Geom. Appl. 6 (996), Department of Mathematics Southeast University, Nanjing P.R. China xlchao@seu.edu.cn