- Email: [email protected]
Complete hypersurfaces with constant mean curvature and finite index in hyperbolic spaces
Acta Mathematica Scientia 2011,31B(1):353–360 http://actams.wipm.ac.cn
COMPLETE HYPERSURFACES WITH CONSTANT MEAN CURVATURE AND FINITE INDEX IN HYPERBOLIC SPACES∗ Deng Qintao (
)
Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China Laboratory of Nonlinear Analysis, and School of Mathematics and Statistics, Huazhong Normal University, Wuhan 430079, China E-mail: [email protected]
Abstract In this article, we prove that any complete finite index hypersurface in the hyperbolic space H4 (−1)(H5 (−1)) with constant mean curvature H satisfying H 2 > 64 63 (H 2 > 175 respectively) must be compact. Specially, we verify that any complete and 148 stable hypersurface in the hyperbolic space H4 (−1) (resp. H5 (−1)) with constant mean (resp. H 2 > 175 ) must be compact. It shows that there curvature H satisfying H 2 > 64 63 148 is no manifold satisfying the conditions of some theorems in [7, 9]. Key words k-weighted bi-Ricci curvature; finite index; constant mean curvature 2000 MR Subject Classification
1
53C40; 53C42
Introduction
Let M n be a complete hypersurface with constant mean curvature H in N n+1 . Throughout ¯ Ric, K; ¯ ∇, Ric, K, A, |A|, and v to denote the connection, Ricci curvature, this article, we use ∇, sectional curvature of N ; and the connection, Ricci curvature, sectional curvature, second fundamental form, and the norm of second fundamental form, the unit normal field of M , respectively. Definition 1.1 [4] Let x : M n −→ N n+1 be an immersion with constant mean curvature H. The stability operator is given by L = + Ric(v) + |A|2 .
(1.1)
Given a relatively compact domain Ω ⊂ M , denote the number of negative eigenvalues of the operator L by Ind(Ω) for the Dirichlet problem on Ω. The index of the immersion is defined by Ind(M ) := sup{Ind(Ω)| Ω ⊂ M, Ω is relatively compact}. ∗ Received
(1.2)
April 23, 2009; revised October 20, 2009. Research was supported by NSFC (10901067) and partially supported by NSFC (10801058) and Hubei Key Laboratory of Mathematical Sciences
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Definition 1.2 An immersion x : M n −→ N n+1 with constant mean curvature H (H = 0) is called stable if |∇f |2 − (Ric(v) + |A|2 )f 2 ≥ 0 (1.3) M
holds for any f ∈
C0∞ (M )
satisfying
f = 0.
(1.4)
M
In this article, we will study stable or finite index constant mean curvature hypersurfaces. From the definitions, we know that stable constant mean curvature hypersurfaces are of finite index. A.M. da Silveira [15] studied constant mean curvature surfaces in H3 (−1) and proved that Theorem 1.3 Any complete finite index surface in the hyperbolic space H3 (−1) with constant mean curvature H satisfying H 2 > 1 must be compact. Xu Cheng [8] generalized this result to H4 (−1) and H5 (−1). Actually, she proved that Theorem 1.4 Any complete finite index hypersurface in the hyperbolic space H4 (−1) 7 2 (resp. H5 (−1)) with constant mean curvature H satisfying H 2 > 10 9 (resp. H > 4 ) must be compact. In this article, we would improve Theorem 1.4 to the following. Theorem 1.5 Any complete finite index hypersurface in the hyperbolic space H4 (−1) 175 2 (resp. H5 (−1)) with constant mean curvature H satisfying H 2 > 64 63 (resp. H > 148 ) must be compact. Theorem 1.6 In [10], L.F. Cheung, M. Do Carmo, and W. Santos proved that any complete finite index hypersurface in Hn+1 (−1) with constant mean curvature H > 1 and |φ|n < ∞ is compact, where φ = A−Hg. Theorem 1.5 can be viewed as a partial generalized M version of this result. We don’t know whether the conclusion holds if the request of H is replaced by H > 1. In 2005, L.F. Cheung and Detang Zhou [11] proved that all complete stable hypersurfaces n+1 in H (−1) (n = 3, 4, 5) with constant mean curvature H > 1 and M |φ|2 < ∞ are compact geodesic spheres. When n = 3, 4, we partially improve this result and prove that Theorem 1.7 All complete stable hypersurfaces in H4 (−1) (resp. H5 (−1)) with constant 175 2 mean curvature H satisfying H 2 > 64 63 (resp. H > 148 ) are compact geodesic spheres. In 2000, Xu Cheng [7] proved that Theorem 1.8 Let M n (n = 3 or 4) be a complete noncompact stongly stable hypersurface in Hn+1 (−1) with constant mean curvature H satisfying H2 ≥
10 7 when n = 3 or H 2 ≥ when n = 4, 9 4
then, there exists no L2 harmonic 1-form on M . In 2006, Xu Cheng, L.F. Cheung & Detang Zhou [9] proved that Theorem 1.9 Any complete noncompact stable hypersurface in H4 (−1) (resp. H5 (−1)) 7 2 with constant mean curvature H satisfying H 2 ≥ 10 9 (resp. H ≥ 4 ) has only one end. But, Theorem 1.7 shows that there is no hypersurface satisfying the conditions of Theorem 1.8 or Theorem 1.9.
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In Section 2, we would introduce the concept of k-weighted bi-Ricci curvature which generalizes the notion of bi-Ricci curvature introduced by Ying Shen and Rugang Ye [16]. Then, we establish an important proposition (see Proposition 2.4), which allows us to improve the result of Xu Cheng. The method mainly follows from Xu Cheng [8], Ying Shen and Ruguang Ye [16], and D. Fischer-Colbrie [12].
2
Proofs of the Theorems First of all, we establish an algebraic lemma: Lemma 2.1 Let A = (aij ) be an n × n real symmetric matrix with TrA = nH. Then, k A 2 + nHa11 −
n
a21i ≥
i=1
for any k >
n−1 n ,
where A 2 =
a11 = −
n i,j=1
4k 2 − n + 1 n2 H 2 4((k − 1)n + 1)
a2ij , and the equality holds if and only if
n − 1 − 2k 2k − 1 nH, aii = nH (∀ i = 1) 2((k − 1)n + 1) 2((k − 1)n + 1)
and aij = 0 for any i = j. Proof Assume n > 1, because it is trivial for n = 1. By a direct computation, we have k A 2 + nHa11 − =k
n
n
a21i
i=1
a2ij + nHa11 −
i,j=1
n
a21i
i=1
≥ ka211 + k(a222 + · · · + a2nn ) + 2k
n i=2
≥ (k − 1)a211 +
a21i + nHa11 −
n
a21i
i=1
n
k (a22 + · · · + ann )2 + (2k − 1) a21i + nHa11 n−1 i=2
k (nH − a11 )2 + nHa11 ≥ (k − 1)a211 + n−1 nk 2k k 2 − 1 a11 + 1 − nHa11 + n2 H 2 = n−1 n−1 n−1 2 nk k 2k 4 n−1 − 1 n−1 − 1 − n−1 n2 H 2 ≥ nk 4 n−1 − 1 =
4k 2 − n + 1 n2 H 2 . 4((k − 1)n + 1)
Thus, the equality in Lemma 2.1 holds if and only if all of the above equalities hold, that is, 1 − 2k a11 = − n−1 nH, a22 = · · · = ann and aij = 0, ∀ i = j. nk 2 n−1 −1
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As TrA = nH, we have a11 = −
n − 1 − 2k 2k − 1 nH, aii = nH (∀ i = 1). 2((k − 1)n + 1) 2((k − 1)n + 1)
Definition 2.2 Given an (n+1)-dimensional Riemannian manifold N n+1 , and two orthonormal tangent vectors u, v, the k-weighted bi-Ricci curvature in the directions (u, v) is defined by ¯ k-bi-Ric(u, v) := Ric(u) + kRic(v) − K(u, v). Remark 2.3 (i) It is clear that 1-weighted bi-Ricci curvature is just the bi-Ricci curvature introduced by Ying Shen and Ruguang Ye [16]. (ii) Generally, the k-weighted bi-Ricci curvature is not symmetric if k = 1. Proposition 2.4 Let N n+1 (n = 3, 4, 5) be a complete (n + 1)-dimensional manifold and M be a complete hypersurface in N n+1 with finite index and constant mean curvature H. If
4 for some k ∈ n−1 n , n−1 ,
4k 2 − n + 1 n2 H 2 , τ := inf k-bi-Ric(w, v)|w ∈ Tp1 M, p ∈ M > − 4((k − 1)n + 1) then, M must be compact, where v denotes the unit normal field on M . Proof We basically follow the arguments as in [8, 16]. Suppose that, on the contrary, M is noncompact. Then, there exists a positive function u on M , such that Δu + (|A|2 + Ric(v))u = 0 on M \ Ω
(2.1)
because M has finite index, where Ω is some compact subset of M [12]. Denote the original metric on M by ds2 . Let the conformal metric be d˜ s2 = u2k ds2 ,
4 where k ∈ n−1 n , n−1 is a constant. Similar to [16] and [8], we can obtain a minimizing geodesic γ˜(s) : [0, ∞) −→ M \ Ω in the metric d˜ s2 , where the parameter s is arc length in the metric ds2 . This is a contradiction to the following claim, which completes the proof of the proposition. Claim There exists a positive constant C(n, k, H, τ ), such that any minimized geodesic in M \ Ω in the metric d˜ s2 has length no greater than C(n, k, H, τ ) in the metric ds2 . Proof of the Claim Assume that c˜(s)(0 ≤ s ≤ l) is any minimized geodesic in M \ Ω, where s is arc length in the metric ds2 . Choose an orthonormal frame {ei }, i = 1, · · · , n of M d along c˜ so that e1 = ds . Then, {e˜i = u−k ei } is an orthonormal frame in the metric d˜ s2 and d e˜1 = d˜ s. ˜ Ric, and K ˜ to denote the connection, Ricci curvature, and In the following, we use ∇, 2 sectional curvature of M in the metric d˜ s , respectively. By the second variation of arc length, we have (n − 1)
˜l 0
dϕ d˜ s
2
d˜ s≥ 0
˜ l
e˜1 )d˜ ϕ2 Ric( s,
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which is equivalent to
l
(n − 1)
u−k
0
dϕ ds
2
l
ds ≥
e˜1 )ds, uk ϕ2 Ric(
(2.2)
0
where ϕ is any smooth function along c˜ with ϕ(0) = ϕ(˜l) = 0 and ˜l is the length of c˜ in the metric d˜ s2 . From the curvature transformation formula for the conformal metric change [15], we have 2 2 d log u Ric(e˜1 ) = Ric(e˜1 ) − (n − 2)kHess(log u)(e˜1 , e˜1 ) + (n − 2)k d˜ s 2 d
− kΔ(log u) + (n − 2)k 2 |∇ log u|2 d˜ s 2 d d log u log u = u−2k Ric(e1 ) − (n − 2)k − ∇ d ds ds ds2 2
−2k 2 d log u 2 2 (n − 2)k +u − kΔ(log u) + (n − 2)k |∇ log u| ds d2 log u −2k Ric(e1 ) − (n − 2)k − kΔ(log u) =u ds2 2 d log u d log u − k |∇ log u|2 − ∇d +(n − 2)ku−2k ds ds ds and
˜ d d = ∇ d d + 2k d log u d − k∇ log u. ∇ ds ds ds ds ds ds
(2.3)
(2.4)
As c˜ is minimizing in the metric d˜ s2 , we have ˜ d d = 0. ∇ d˜ s d˜ s
(2.5)
By a direct computation, we have ˜ d d =∇ ˜ −k d u−k d = −ku−(2k+1) du d + u−2k ∇ ˜ d d. ∇ u d˜ s d˜ ds ds ds s ds ds ds
(2.6)
From (2.4), (2.5), and (2.6), we have
d ∇d ds ds
d log u 2 . log u = k |∇ log u| − ds 2
(2.7)
Δu + |∇ log u|2 = |A|2 + Ric(v) + |∇ log u|2 u
(2.8)
From (2.1), we have −Δ(log u) = − and recall the Gauss equation: ¯ 1 , v) + nh11 − Ric(e1 ) = Ric(e1 ) − K(e
n j=1
h21j .
(2.9)
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Substituting (2.7), (2.8), and (2.9) into (2.3), we obtain e˜1 ) = k-bi-Ric(e1 , v) + k|A|2 + nh11 − u2k Ric(
n
h21j − (n − 2)k
j=1
d2 log u + k|∇ log u|2 (2.10) ds2
Substituting (2.10) into (2.2) and replace ϕ by f uk/2 , we have 2 l 2 l df k 2 l 2 d log u df d log u (n − 1) ds + f ds + k f ds 4 0 ds ds ds 0 0 ⎛ ⎞ l n ≥ f 2 ⎝k-bi-Ric(e1 , v) + k|A|2 + nh11 − h21j ⎠ ds 0
l
2
j=1
l
l
d log u f2 ds + k |∇ log u|2 ds −(n − 2)k ds2 0 0 ⎛ ⎞ l n f 2 ⎝k-bi-Ric(e1 , v) + k|A|2 + nh11 − h21j ⎠ ds = 0
l
l
j=1
df d log u +k f |∇ log u|2 ds (Integrating by parts) +2(n − 2)k ds ds 0 0 ⎛ ⎞ l n f 2 ⎝k-bi-Ric(e1 , v) + k|A|2 + nh11 − h21j ⎠ ds ≥ 0
+2(n − 2)k 0
df d log u +k f ds ds
l
f 0
2
j=1
d log u ds
2 ds.
(2.11)
Rearrange (2.11), we obtain l
df ds
2
(n − 1) ds ⎛0 ⎞ l n f 2 ⎝k-bi-Ric(e1 , v) + k|A|2 + nh11 − h21j ⎠ ds ≥ 0
j=1
l 2 l d log u df d log u n−1 k . f2 ds + (n − 3)k f +k 1 − 4 ds ds ds 0 0 By the H¨ older inequality, we have l df d log u (n − 3)k f ds ds 0 l 2 l 2 2 k d log u df (n − 3) n−1 k f2 ds + ds . ≥ − k 1− 4 ds 4 − (n − 1)k ds 0 0
(2.12)
(2.13)
Combining (2.12) and (2.13) and using Lemma 2.1, we have l c(n, k) 0
df ds
where c(n, k) = n − 1 +
2
ds ≥
l
f k-bi-Ric(e1 , v) + 2
0 (n−3)2 k 4−(n−1)k
> 0.
4k 2 − n + 1 2 2 n H ds 4((k − 1)n + 1)
(2.14)
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As k-bi-Ric(e1 , v) ≥ τ , we have l c(n, k) 0
df ds
2
ds ≥
l
f2 τ +
0
4k 2 − n + 1 n2 H 2 ds. 4((k − 1)n + 1)
(2.15)
Choose f = sin πs l , 0 ≤ s ≤ l, we obtain l≤π
c(n, k) 4k2 −n+1 2 2 4((k−1)n+1) n H
τ+
:= C(n, k, H, τ ).
Theorem 2.5 Let N n+1 (n = 3, 4) be a complete (n+1)-dimensional manifold with sec˜ ≥ −τ, τ ≥ 0, and M be a complete hypersurface in N n+1 with finite index tional curvature K and constant mean curvature H. If H satisfies H2 >
64 175 τ when n = 3; H 2 > τ when n = 4, 63 148
then M must be compact. Proof It is checked that k-bi-Ric ≥ −((k + 1)n − 1))τ. By Proposition 2.4, we conclude that M must be compact when 4k 2 − n + 1 n2 H 2 4((k − 1)n + 1)
(2.16)
4((k + 1)n − 1)((k − 1)n + 1) τ. (4k 2 − n + 1)n2
(2.17)
−((k + 1)n − 1)τ > − for some k ∈
n−1 n
,
4 n−1
. While (2.16) holds if
H2 > k∈
√
inf
n−1 2
,
4 n−1
By a direct computation, we verify that (2.17) is equivalent to H2 >
64 175 τ when n = 3; H 2 > τ when n = 4. 63 148
Specially, we have Corollary 2.6 Any complete finite index hypersurface in the hyperbolic space H4 (−1) 175 2 (resp. H5 (−1)) with constant mean curvature H satisfying H 2 > 64 63 (resp. H > 148 ) must be compact. ¯ = −1, Theorem 2.5 follows directly if we take τ = −1. Proof As K Corollary 2.7 Any complete stable hypersurface M in H4 (−1) (resp. H5 (−1)) with 175 2 constant mean curvature H satisfying H 2 > 64 63 (resp. H > 148 ) is a compact geodesic sphere. Proof As stability of M implies that M has finite index, we verify that M is compact according to Corollary 2.6. Then, we conclude that M |φ|2 < ∞ and M is a compact geodesic sphere by the result of L.F. Cheung and Detang Zhou[11], which says that all complete stable hypersurface in Hn+1 (−1) (n = 3, 4, 5) with constant mean curvature H > 1 and M |φ|2 < ∞ are compact geodesic spheres. Remark 2.8 We don’t know whether the dimension condition is sharp.
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Acknowledgments The author would like to thank Professor Li Gongbao for his constant encouragement and help. References [1] Alencar H, Do Carmo M. Hypersurfaces with constant mean curvature in space forms. An Acad Bras Ci, 1994, 66: 265–274 [2] Barbosa J L, Do Carmo M, Eschenburg J. Stability of hypersurfaces of constant mean curvature in Riemannian manifolds. Math Z, 1988, 197(1): 123–138 [3] Berard P, Do Carmo M, Santos W. Complete hypersurfaces with contant mean curvature and finite total curvature. Annals of Global Annalysis and Geometry, 1998, 16: 273–290 [4] Berard P, Santos W. Curvature estimates and stability properties of CMC-submanifolds in space forms. Matematica Contemporanea, 1999, 17: 77–97 [5] Do Carmo M, Zhou D. Erratum to: “Bernstein-type theorems in hypersurfaces with constant mean curvature”. An Acad Brasil Ciˆenc, 2001, 73(3): 333–335 [6] Do Carmo M, Zhou D. Bernstein-type theorems in hypersurfaces with constant mean curvature. An Acad Brasil Ciˆenc, 2000, 72(3): 301–310 [7] Cheng X. L2 Harmonic Forms and Stability of Hypersurfaces with Constant Mean Curvature. Bol Soc Bras Mat, 2000, 31(2): 225–239 [8] Cheng X. On constant mean curvature hypersurfaces with finite index. Arch Math, 2006, 86: 365–374 [9] Cheng X, Cheung L F, Zhou D. The Structure of Stable Constant Mean Curvature Hypersurfaces. arXiv:math/0602007v1 [math.DG]. [10] Cheung L F, Do Carmo M, Santos W. On the compactness of CMC-hupersurfaces with finite total curvature. Arch Math, 1999, 73(3): 216–222 [11] Cheung L F, Zhou D. Stable constant mean curvature hypersurfaces in Rn+1 and Hn+1 (−1). Bull Braz Math Soc (NS), 2005, 36(1): 99–114 [12] Fischer-Colbrie D. On complete minimal surfaces with finite Morse index in three manifolds. Invent Math, 1985, 82: 121–132 [13] Li P, Wang J P. Minimal hypersurfaces with finite index. Math Res Lett, 2002, 9: 95–103 [14] Lopez F, Ros A. Complete minimal surfaces with index one and stable constant mean curvature surfaces. Comm Math Helvetici, 1989, 64: 34–43 [15] da Silveira A M. Stability of Complete Noncompact Surfaces with Constant Mean Curvature. Math Ann, 1987, 277(4): 629–638 [16] Shen Y, Ye R. On stable minimal surfaces in manifolds of positive Bi-Ricci curvature. Duke Math J, 1996, 85: 109–116 [17] Simons J. Minimal varieties in Riemannian manifolds. Ann Math, 1968, 88: 62–105 [18] Schoen R, Yau S T. Lectures on differential geometry. Conference Proceedings and Lecture Note in Goemetry and Topology I, Combrige, MA, 1994
COMPLETE HYPERSURFACES WITH CONSTANT MEAN CURVATURE AND FINITE INDEX IN HYPERBOLIC SPACES∗ Deng Qintao (
)
Department of Mathematics, Sun Yat-Sen University, Guangzhou 510275, China Laboratory of Nonlinear Analysis, and School of Mathematics and Statistics, Huazhong Normal University, Wuhan 430079, China E-mail: [email protected]
Abstract In this article, we prove that any complete finite index hypersurface in the hyperbolic space H4 (−1)(H5 (−1)) with constant mean curvature H satisfying H 2 > 64 63 (H 2 > 175 respectively) must be compact. Specially, we verify that any complete and 148 stable hypersurface in the hyperbolic space H4 (−1) (resp. H5 (−1)) with constant mean (resp. H 2 > 175 ) must be compact. It shows that there curvature H satisfying H 2 > 64 63 148 is no manifold satisfying the conditions of some theorems in [7, 9]. Key words k-weighted bi-Ricci curvature; finite index; constant mean curvature 2000 MR Subject Classification
1
53C40; 53C42
Introduction
Let M n be a complete hypersurface with constant mean curvature H in N n+1 . Throughout ¯ Ric, K; ¯ ∇, Ric, K, A, |A|, and v to denote the connection, Ricci curvature, this article, we use ∇, sectional curvature of N ; and the connection, Ricci curvature, sectional curvature, second fundamental form, and the norm of second fundamental form, the unit normal field of M , respectively. Definition 1.1 [4] Let x : M n −→ N n+1 be an immersion with constant mean curvature H. The stability operator is given by L = + Ric(v) + |A|2 .
(1.1)
Given a relatively compact domain Ω ⊂ M , denote the number of negative eigenvalues of the operator L by Ind(Ω) for the Dirichlet problem on Ω. The index of the immersion is defined by Ind(M ) := sup{Ind(Ω)| Ω ⊂ M, Ω is relatively compact}. ∗ Received
(1.2)
April 23, 2009; revised October 20, 2009. Research was supported by NSFC (10901067) and partially supported by NSFC (10801058) and Hubei Key Laboratory of Mathematical Sciences
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Definition 1.2 An immersion x : M n −→ N n+1 with constant mean curvature H (H = 0) is called stable if |∇f |2 − (Ric(v) + |A|2 )f 2 ≥ 0 (1.3) M
holds for any f ∈
C0∞ (M )
satisfying
f = 0.
(1.4)
M
In this article, we will study stable or finite index constant mean curvature hypersurfaces. From the definitions, we know that stable constant mean curvature hypersurfaces are of finite index. A.M. da Silveira [15] studied constant mean curvature surfaces in H3 (−1) and proved that Theorem 1.3 Any complete finite index surface in the hyperbolic space H3 (−1) with constant mean curvature H satisfying H 2 > 1 must be compact. Xu Cheng [8] generalized this result to H4 (−1) and H5 (−1). Actually, she proved that Theorem 1.4 Any complete finite index hypersurface in the hyperbolic space H4 (−1) 7 2 (resp. H5 (−1)) with constant mean curvature H satisfying H 2 > 10 9 (resp. H > 4 ) must be compact. In this article, we would improve Theorem 1.4 to the following. Theorem 1.5 Any complete finite index hypersurface in the hyperbolic space H4 (−1) 175 2 (resp. H5 (−1)) with constant mean curvature H satisfying H 2 > 64 63 (resp. H > 148 ) must be compact. Theorem 1.6 In [10], L.F. Cheung, M. Do Carmo, and W. Santos proved that any complete finite index hypersurface in Hn+1 (−1) with constant mean curvature H > 1 and |φ|n < ∞ is compact, where φ = A−Hg. Theorem 1.5 can be viewed as a partial generalized M version of this result. We don’t know whether the conclusion holds if the request of H is replaced by H > 1. In 2005, L.F. Cheung and Detang Zhou [11] proved that all complete stable hypersurfaces n+1 in H (−1) (n = 3, 4, 5) with constant mean curvature H > 1 and M |φ|2 < ∞ are compact geodesic spheres. When n = 3, 4, we partially improve this result and prove that Theorem 1.7 All complete stable hypersurfaces in H4 (−1) (resp. H5 (−1)) with constant 175 2 mean curvature H satisfying H 2 > 64 63 (resp. H > 148 ) are compact geodesic spheres. In 2000, Xu Cheng [7] proved that Theorem 1.8 Let M n (n = 3 or 4) be a complete noncompact stongly stable hypersurface in Hn+1 (−1) with constant mean curvature H satisfying H2 ≥
10 7 when n = 3 or H 2 ≥ when n = 4, 9 4
then, there exists no L2 harmonic 1-form on M . In 2006, Xu Cheng, L.F. Cheung & Detang Zhou [9] proved that Theorem 1.9 Any complete noncompact stable hypersurface in H4 (−1) (resp. H5 (−1)) 7 2 with constant mean curvature H satisfying H 2 ≥ 10 9 (resp. H ≥ 4 ) has only one end. But, Theorem 1.7 shows that there is no hypersurface satisfying the conditions of Theorem 1.8 or Theorem 1.9.
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In Section 2, we would introduce the concept of k-weighted bi-Ricci curvature which generalizes the notion of bi-Ricci curvature introduced by Ying Shen and Rugang Ye [16]. Then, we establish an important proposition (see Proposition 2.4), which allows us to improve the result of Xu Cheng. The method mainly follows from Xu Cheng [8], Ying Shen and Ruguang Ye [16], and D. Fischer-Colbrie [12].
2
Proofs of the Theorems First of all, we establish an algebraic lemma: Lemma 2.1 Let A = (aij ) be an n × n real symmetric matrix with TrA = nH. Then, k A 2 + nHa11 −
n
a21i ≥
i=1
for any k >
n−1 n ,
where A 2 =
a11 = −
n i,j=1
4k 2 − n + 1 n2 H 2 4((k − 1)n + 1)
a2ij , and the equality holds if and only if
n − 1 − 2k 2k − 1 nH, aii = nH (∀ i = 1) 2((k − 1)n + 1) 2((k − 1)n + 1)
and aij = 0 for any i = j. Proof Assume n > 1, because it is trivial for n = 1. By a direct computation, we have k A 2 + nHa11 − =k
n
n
a21i
i=1
a2ij + nHa11 −
i,j=1
n
a21i
i=1
≥ ka211 + k(a222 + · · · + a2nn ) + 2k
n i=2
≥ (k − 1)a211 +
a21i + nHa11 −
n
a21i
i=1
n
k (a22 + · · · + ann )2 + (2k − 1) a21i + nHa11 n−1 i=2
k (nH − a11 )2 + nHa11 ≥ (k − 1)a211 + n−1 nk 2k k 2 − 1 a11 + 1 − nHa11 + n2 H 2 = n−1 n−1 n−1 2 nk k 2k 4 n−1 − 1 n−1 − 1 − n−1 n2 H 2 ≥ nk 4 n−1 − 1 =
4k 2 − n + 1 n2 H 2 . 4((k − 1)n + 1)
Thus, the equality in Lemma 2.1 holds if and only if all of the above equalities hold, that is, 1 − 2k a11 = − n−1 nH, a22 = · · · = ann and aij = 0, ∀ i = j. nk 2 n−1 −1
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As TrA = nH, we have a11 = −
n − 1 − 2k 2k − 1 nH, aii = nH (∀ i = 1). 2((k − 1)n + 1) 2((k − 1)n + 1)
Definition 2.2 Given an (n+1)-dimensional Riemannian manifold N n+1 , and two orthonormal tangent vectors u, v, the k-weighted bi-Ricci curvature in the directions (u, v) is defined by ¯ k-bi-Ric(u, v) := Ric(u) + kRic(v) − K(u, v). Remark 2.3 (i) It is clear that 1-weighted bi-Ricci curvature is just the bi-Ricci curvature introduced by Ying Shen and Ruguang Ye [16]. (ii) Generally, the k-weighted bi-Ricci curvature is not symmetric if k = 1. Proposition 2.4 Let N n+1 (n = 3, 4, 5) be a complete (n + 1)-dimensional manifold and M be a complete hypersurface in N n+1 with finite index and constant mean curvature H. If
4 for some k ∈ n−1 n , n−1 ,
4k 2 − n + 1 n2 H 2 , τ := inf k-bi-Ric(w, v)|w ∈ Tp1 M, p ∈ M > − 4((k − 1)n + 1) then, M must be compact, where v denotes the unit normal field on M . Proof We basically follow the arguments as in [8, 16]. Suppose that, on the contrary, M is noncompact. Then, there exists a positive function u on M , such that Δu + (|A|2 + Ric(v))u = 0 on M \ Ω
(2.1)
because M has finite index, where Ω is some compact subset of M [12]. Denote the original metric on M by ds2 . Let the conformal metric be d˜ s2 = u2k ds2 ,
4 where k ∈ n−1 n , n−1 is a constant. Similar to [16] and [8], we can obtain a minimizing geodesic γ˜(s) : [0, ∞) −→ M \ Ω in the metric d˜ s2 , where the parameter s is arc length in the metric ds2 . This is a contradiction to the following claim, which completes the proof of the proposition. Claim There exists a positive constant C(n, k, H, τ ), such that any minimized geodesic in M \ Ω in the metric d˜ s2 has length no greater than C(n, k, H, τ ) in the metric ds2 . Proof of the Claim Assume that c˜(s)(0 ≤ s ≤ l) is any minimized geodesic in M \ Ω, where s is arc length in the metric ds2 . Choose an orthonormal frame {ei }, i = 1, · · · , n of M d along c˜ so that e1 = ds . Then, {e˜i = u−k ei } is an orthonormal frame in the metric d˜ s2 and d e˜1 = d˜ s. ˜ Ric, and K ˜ to denote the connection, Ricci curvature, and In the following, we use ∇, 2 sectional curvature of M in the metric d˜ s , respectively. By the second variation of arc length, we have (n − 1)
˜l 0
dϕ d˜ s
2
d˜ s≥ 0
˜ l
e˜1 )d˜ ϕ2 Ric( s,
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which is equivalent to
l
(n − 1)
u−k
0
dϕ ds
2
l
ds ≥
e˜1 )ds, uk ϕ2 Ric(
(2.2)
0
where ϕ is any smooth function along c˜ with ϕ(0) = ϕ(˜l) = 0 and ˜l is the length of c˜ in the metric d˜ s2 . From the curvature transformation formula for the conformal metric change [15], we have 2 2 d log u Ric(e˜1 ) = Ric(e˜1 ) − (n − 2)kHess(log u)(e˜1 , e˜1 ) + (n − 2)k d˜ s 2 d
− kΔ(log u) + (n − 2)k 2 |∇ log u|2 d˜ s 2 d d log u log u = u−2k Ric(e1 ) − (n − 2)k − ∇ d ds ds ds2 2
−2k 2 d log u 2 2 (n − 2)k +u − kΔ(log u) + (n − 2)k |∇ log u| ds d2 log u −2k Ric(e1 ) − (n − 2)k − kΔ(log u) =u ds2 2 d log u d log u − k |∇ log u|2 − ∇d +(n − 2)ku−2k ds ds ds and
˜ d d = ∇ d d + 2k d log u d − k∇ log u. ∇ ds ds ds ds ds ds
(2.3)
(2.4)
As c˜ is minimizing in the metric d˜ s2 , we have ˜ d d = 0. ∇ d˜ s d˜ s
(2.5)
By a direct computation, we have ˜ d d =∇ ˜ −k d u−k d = −ku−(2k+1) du d + u−2k ∇ ˜ d d. ∇ u d˜ s d˜ ds ds ds s ds ds ds
(2.6)
From (2.4), (2.5), and (2.6), we have
d ∇d ds ds
d log u 2 . log u = k |∇ log u| − ds 2
(2.7)
Δu + |∇ log u|2 = |A|2 + Ric(v) + |∇ log u|2 u
(2.8)
From (2.1), we have −Δ(log u) = − and recall the Gauss equation: ¯ 1 , v) + nh11 − Ric(e1 ) = Ric(e1 ) − K(e
n j=1
h21j .
(2.9)
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Substituting (2.7), (2.8), and (2.9) into (2.3), we obtain e˜1 ) = k-bi-Ric(e1 , v) + k|A|2 + nh11 − u2k Ric(
n
h21j − (n − 2)k
j=1
d2 log u + k|∇ log u|2 (2.10) ds2
Substituting (2.10) into (2.2) and replace ϕ by f uk/2 , we have 2 l 2 l df k 2 l 2 d log u df d log u (n − 1) ds + f ds + k f ds 4 0 ds ds ds 0 0 ⎛ ⎞ l n ≥ f 2 ⎝k-bi-Ric(e1 , v) + k|A|2 + nh11 − h21j ⎠ ds 0
l
2
j=1
l
l
d log u f2 ds + k |∇ log u|2 ds −(n − 2)k ds2 0 0 ⎛ ⎞ l n f 2 ⎝k-bi-Ric(e1 , v) + k|A|2 + nh11 − h21j ⎠ ds = 0
l
l
j=1
df d log u +k f |∇ log u|2 ds (Integrating by parts) +2(n − 2)k ds ds 0 0 ⎛ ⎞ l n f 2 ⎝k-bi-Ric(e1 , v) + k|A|2 + nh11 − h21j ⎠ ds ≥ 0
+2(n − 2)k 0
df d log u +k f ds ds
l
f 0
2
j=1
d log u ds
2 ds.
(2.11)
Rearrange (2.11), we obtain l
df ds
2
(n − 1) ds ⎛0 ⎞ l n f 2 ⎝k-bi-Ric(e1 , v) + k|A|2 + nh11 − h21j ⎠ ds ≥ 0
j=1
l 2 l d log u df d log u n−1 k . f2 ds + (n − 3)k f +k 1 − 4 ds ds ds 0 0 By the H¨ older inequality, we have l df d log u (n − 3)k f ds ds 0 l 2 l 2 2 k d log u df (n − 3) n−1 k f2 ds + ds . ≥ − k 1− 4 ds 4 − (n − 1)k ds 0 0
(2.12)
(2.13)
Combining (2.12) and (2.13) and using Lemma 2.1, we have l c(n, k) 0
df ds
where c(n, k) = n − 1 +
2
ds ≥
l
f k-bi-Ric(e1 , v) + 2
0 (n−3)2 k 4−(n−1)k
> 0.
4k 2 − n + 1 2 2 n H ds 4((k − 1)n + 1)
(2.14)
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As k-bi-Ric(e1 , v) ≥ τ , we have l c(n, k) 0
df ds
2
ds ≥
l
f2 τ +
0
4k 2 − n + 1 n2 H 2 ds. 4((k − 1)n + 1)
(2.15)
Choose f = sin πs l , 0 ≤ s ≤ l, we obtain l≤π
c(n, k) 4k2 −n+1 2 2 4((k−1)n+1) n H
τ+
:= C(n, k, H, τ ).
Theorem 2.5 Let N n+1 (n = 3, 4) be a complete (n+1)-dimensional manifold with sec˜ ≥ −τ, τ ≥ 0, and M be a complete hypersurface in N n+1 with finite index tional curvature K and constant mean curvature H. If H satisfies H2 >
64 175 τ when n = 3; H 2 > τ when n = 4, 63 148
then M must be compact. Proof It is checked that k-bi-Ric ≥ −((k + 1)n − 1))τ. By Proposition 2.4, we conclude that M must be compact when 4k 2 − n + 1 n2 H 2 4((k − 1)n + 1)
(2.16)
4((k + 1)n − 1)((k − 1)n + 1) τ. (4k 2 − n + 1)n2
(2.17)
−((k + 1)n − 1)τ > − for some k ∈
n−1 n
,
4 n−1
. While (2.16) holds if
H2 > k∈
√
inf
n−1 2
,
4 n−1
By a direct computation, we verify that (2.17) is equivalent to H2 >
64 175 τ when n = 3; H 2 > τ when n = 4. 63 148
Specially, we have Corollary 2.6 Any complete finite index hypersurface in the hyperbolic space H4 (−1) 175 2 (resp. H5 (−1)) with constant mean curvature H satisfying H 2 > 64 63 (resp. H > 148 ) must be compact. ¯ = −1, Theorem 2.5 follows directly if we take τ = −1. Proof As K Corollary 2.7 Any complete stable hypersurface M in H4 (−1) (resp. H5 (−1)) with 175 2 constant mean curvature H satisfying H 2 > 64 63 (resp. H > 148 ) is a compact geodesic sphere. Proof As stability of M implies that M has finite index, we verify that M is compact according to Corollary 2.6. Then, we conclude that M |φ|2 < ∞ and M is a compact geodesic sphere by the result of L.F. Cheung and Detang Zhou[11], which says that all complete stable hypersurface in Hn+1 (−1) (n = 3, 4, 5) with constant mean curvature H > 1 and M |φ|2 < ∞ are compact geodesic spheres. Remark 2.8 We don’t know whether the dimension condition is sharp.
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Acknowledgments The author would like to thank Professor Li Gongbao for his constant encouragement and help. References [1] Alencar H, Do Carmo M. Hypersurfaces with constant mean curvature in space forms. An Acad Bras Ci, 1994, 66: 265–274 [2] Barbosa J L, Do Carmo M, Eschenburg J. Stability of hypersurfaces of constant mean curvature in Riemannian manifolds. Math Z, 1988, 197(1): 123–138 [3] Berard P, Do Carmo M, Santos W. Complete hypersurfaces with contant mean curvature and finite total curvature. Annals of Global Annalysis and Geometry, 1998, 16: 273–290 [4] Berard P, Santos W. Curvature estimates and stability properties of CMC-submanifolds in space forms. Matematica Contemporanea, 1999, 17: 77–97 [5] Do Carmo M, Zhou D. Erratum to: “Bernstein-type theorems in hypersurfaces with constant mean curvature”. An Acad Brasil Ciˆenc, 2001, 73(3): 333–335 [6] Do Carmo M, Zhou D. Bernstein-type theorems in hypersurfaces with constant mean curvature. An Acad Brasil Ciˆenc, 2000, 72(3): 301–310 [7] Cheng X. L2 Harmonic Forms and Stability of Hypersurfaces with Constant Mean Curvature. Bol Soc Bras Mat, 2000, 31(2): 225–239 [8] Cheng X. On constant mean curvature hypersurfaces with finite index. Arch Math, 2006, 86: 365–374 [9] Cheng X, Cheung L F, Zhou D. The Structure of Stable Constant Mean Curvature Hypersurfaces. arXiv:math/0602007v1 [math.DG]. [10] Cheung L F, Do Carmo M, Santos W. On the compactness of CMC-hupersurfaces with finite total curvature. Arch Math, 1999, 73(3): 216–222 [11] Cheung L F, Zhou D. Stable constant mean curvature hypersurfaces in Rn+1 and Hn+1 (−1). Bull Braz Math Soc (NS), 2005, 36(1): 99–114 [12] Fischer-Colbrie D. On complete minimal surfaces with finite Morse index in three manifolds. Invent Math, 1985, 82: 121–132 [13] Li P, Wang J P. Minimal hypersurfaces with finite index. Math Res Lett, 2002, 9: 95–103 [14] Lopez F, Ros A. Complete minimal surfaces with index one and stable constant mean curvature surfaces. Comm Math Helvetici, 1989, 64: 34–43 [15] da Silveira A M. Stability of Complete Noncompact Surfaces with Constant Mean Curvature. Math Ann, 1987, 277(4): 629–638 [16] Shen Y, Ye R. On stable minimal surfaces in manifolds of positive Bi-Ricci curvature. Duke Math J, 1996, 85: 109–116 [17] Simons J. Minimal varieties in Riemannian manifolds. Ann Math, 1968, 88: 62–105 [18] Schoen R, Yau S T. Lectures on differential geometry. Conference Proceedings and Lecture Note in Goemetry and Topology I, Combrige, MA, 1994