Characterization of Hyperbolic Cylinders in a Lorentzian Space Form
We give a characterization of the n-dimensional (n ≥ 3) hyperbolic cylinders in a Lorentzian space form. We show that the hyperbolic cylinders are the only complete space-like hypersurfaces in an (n + 1)-dimensional Lorentzian space form M₁ⁿ⁺¹(c) with non-zero constant mean curvature H whose two dis...
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Shu, Sh. Han, Annie Yi 2016-10-03T12:59:13Z 2016-10-03T12:59:13Z 2012 Characterization of Hyperbolic Cylinders in a Lorentzian Space Form / Sh. Shu , Annie Yi Han // Журнал математической физики, анализа, геометрии. — 2012. — Т. 8, № 1. — С. 79-89. — Бібліогр.: 18 назв. — англ. 1812-9471 https://nasplib.isofts.kiev.ua/handle/123456789/106709 We give a characterization of the n-dimensional (n ≥ 3) hyperbolic cylinders in a Lorentzian space form. We show that the hyperbolic cylinders are the only complete space-like hypersurfaces in an (n + 1)-dimensional Lorentzian space form M₁ⁿ⁺¹(c) with non-zero constant mean curvature H whose two distinct principal curvatures λ and μ satisfy inf(λ - μ)² > 0 for c ≤ 0 or inf(λ - μ)² > 0, H² ≥ c, for c > 0, where λ is of multiplicity n - 1 and μ of multiplicity 1 and λ < μ. Дается характеризация n-мерных (n ≥ 3) гиперболических цилиндров в лоренцевой пространственной форме. Показано, что гиперболические цилиндры являются единственными полными пространственноподобными гиперповерхностями в (n + 1)-мерной лоренцевой пространственной форме M₁ⁿ⁺¹(c) с ненулевой постоянной средней кривизны H, у которых две различные главные кривизны λ и μ удовлетворяют inf(λ - μ)² > 0 при c ≤ 0 или inf(λ - μ)² > 0, H² ≥ c, при c > 0, где λ имеет порядок n - 1, а μ порядок 1 и λ < μ. The authors would like to thank the referee for his/her many valuable suggestions and comments made that significantly improved the paper. en Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України Журнал математической физики, анализа, геометрии Characterization of Hyperbolic Cylinders in a Lorentzian Space Form Article published earlier |
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Characterization of Hyperbolic Cylinders in a Lorentzian Space Form |
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Characterization of Hyperbolic Cylinders in a Lorentzian Space Form Shu, Sh. Han, Annie Yi |
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Characterization of Hyperbolic Cylinders in a Lorentzian Space Form |
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Characterization of Hyperbolic Cylinders in a Lorentzian Space Form |
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Characterization of Hyperbolic Cylinders in a Lorentzian Space Form |
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Characterization of Hyperbolic Cylinders in a Lorentzian Space Form |
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characterization of hyperbolic cylinders in a lorentzian space form |
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Shu, Sh. Han, Annie Yi |
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We give a characterization of the n-dimensional (n ≥ 3) hyperbolic cylinders in a Lorentzian space form. We show that the hyperbolic cylinders are the only complete space-like hypersurfaces in an (n + 1)-dimensional Lorentzian space form M₁ⁿ⁺¹(c) with non-zero constant mean curvature H whose two distinct principal curvatures λ and μ satisfy inf(λ - μ)² > 0 for c ≤ 0 or inf(λ - μ)² > 0, H² ≥ c, for c > 0, where λ is of multiplicity n - 1 and μ of multiplicity 1 and λ < μ.
Дается характеризация n-мерных (n ≥ 3) гиперболических цилиндров в лоренцевой пространственной форме. Показано, что гиперболические цилиндры являются единственными полными пространственноподобными гиперповерхностями в (n + 1)-мерной лоренцевой пространственной форме M₁ⁿ⁺¹(c) с ненулевой постоянной средней кривизны H, у которых две различные главные кривизны λ и μ удовлетворяют inf(λ - μ)² > 0 при c ≤ 0 или inf(λ - μ)² > 0, H² ≥ c, при c > 0, где λ имеет порядок n - 1, а μ порядок 1 и λ < μ.
|
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1812-9471 |
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https://nasplib.isofts.kiev.ua/handle/123456789/106709 |
| citation_txt |
Characterization of Hyperbolic Cylinders in a Lorentzian Space Form / Sh. Shu , Annie Yi Han // Журнал математической физики, анализа, геометрии. — 2012. — Т. 8, № 1. — С. 79-89. — Бібліогр.: 18 назв. — англ. |
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2025-11-24T16:28:14Z |
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2025-11-24T16:28:14Z |
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1850485949778100224 |
| fulltext |
Journal of Mathematical Physics, Analysis, Geometry
2012, v. 8, No. 1, pp. 79–89
Characterization of Hyperbolic Cylinders
in a Lorentzian Space Form
Shichang Shu
Department of Mathematics, Xianyang Normal University
Xianyang 712000, Shaanxi, P. R. China
E-mail: shushichang@126.com
Annie Yi Han
Department of Mathematics, Borough of Manhattan Community College
CUNY 10007 N. Y. USA
E-mail: DrYHan@nyc.rr.com
Received January 8, 2009
We give a characterization of the n-dimensional (n ≥ 3) hyperbolic cylin-
ders in a Lorentzian space form. We show that the hyperbolic cylinders
are the only complete space-like hypersurfaces in an (n + 1)-dimensional
Lorentzian space form Mn+1
1 (c) with non-zero constant mean curvature H
whose two distinct principal curvatures λ and µ satisfy inf(λ − µ)2 > 0 for
c ≤ 0 or inf(λ− µ)2 > 0, H2 ≥ c, for c > 0, where λ is of multiplicity n− 1
and µ of multiplicity 1 and λ < µ.
Key words: space-like hypersurface, Lorentzian space form, mean curva-
ture, principal curvature, hyperbolic cylinder.
Mathematics Subject Classification 2000: 53C42, 53A10.
1. Introduction
By an (n + 1)-dimensional Lorentzian space form Mn+1
1 (c) we mean a Min-
kowski space Rn+1
1 , a de Sitter space Sn+1
1 (c) or an anti-de Sitter space Hn+1
1 (c),
according to c > 0, c = 0 or c < 0, respectively. That is, a Lorentzian space form
Mn+1
1 (c) is a complete simply connected (n+1)-dimensional Lorentzian manifold
with constant curvature c. A hypersurface in a Lorentzian manifold is said to be
space-like if the induced metric on the hypersurface is positive definite.
Project supported by NSF of Shaanxi Province (SJ08A31) and NSF of Shaanxi Educational
Committee (11JK0479).
c© Shichang Shu and Annie Yi Han, 2012
Shichang Shu and Annie Yi Han
In connection with the negative settlement of the Bernstein problem due to
Calabi [1] and Cheng-Yau [2], Choquet-Bruhat et al. [3] proved the following
theorem:
Theorem 1.1 ([3]). Let M be a complete space-like hypersurface in an (n+1)-
dimensional Lorentzian space form Mn+1
1 (c), c ≥ 0. If M is maximal, then it is
totally geodesic.
T. Ishihara [4] also proved the following well-known result:
Theorem 1.2 ([4]). Let M be an n-dimensional (n ≥ 2) complete maximal
space-like hypersurface in anti-de Sitter space Hn+1
1 (−1), then
S ≤ n, (1.1)
and S = n if and only if M = Hm(− n
m)×Hn−m(− n
n−m),(1 ≤ m ≤ n− 1), where
S denotes the square of the norm of the second fundamental form of M .
Recently, L. Cao and G. Wei [5] gave a new characterization of hyperbolic
cylinders in anti-de Sitter space Hn+1
1 (−1) as follows:
Theorem 1.3 ([5]). Let M be an n-dimensional (n ≥ 3) complete maximal
space-like hypersurface with two distinct principal curvatures λ and µ in anti-de
Sitter space Hn+1
1 (−1). If inf(λ−µ)2 > 0, then M = Hm(− n
m)×Hn−m(− n
n−m),
(1 ≤ m ≤ n− 1).
As a generalization of Theorem 1.1, the complete space-like hypersurfaces
with constant mean curvature in a Lorentz manifold were studied by many math-
ematicians, see, for instance, ([6–12]). We should note that two types of the
well-known standard models of complete space-like hypersurfaces with non-zero
constant mean curvature in an (n+1)-dimensional Lorentzian space form Mn+1
1 (c)
are the totally umbilical space-like hypersurfaces and the following product man-
ifolds:
Hk(c1)× Sn−k(c2) in Sn+1
1 (c), (
1
c1
+
1
c2
=
1
c
, c1 < 0, c2 > 0),
Hk(c1)×Rn−k in Rn+1
1 , (c1 < 0, c = c2 = 0),
Hk(c1)×Hn−k(c2) in Hn+1
1 (c), (
1
c1
+
1
c2
=
1
c
, c1 < 0, c2 < 0),
where k = 1, . . . , n−1. These three product hypersurfaces are respectively called
the hyperbolic cylinders in Sn+1
1 (c), Rn+1
1 or Hn+1
1 (c). From U-H. Ki et al. [9], we
know that the hyperbolic cylinder H1(c1)×Sn−1(c2) in Sn+1
1 (c) has two distinct
principal curvatures
√
c− c1 with multiplicity 1 and
√
c− c2 with multiplicity
n− 1; the hyperbolic cylinder H1(c1)×Rn−1 in Rn+1
1 has two distinct principal
80 Journal of Mathematical Physics, Analysis, Geometry, 2012, v. 8, No. 1
Characterization of Hyperbolic Cylinders in a Lorentzian Space Form
curvatures
√−c1 with multiplicity 1 and 0 with multiplicity n−1; the hyperbolic
cylinder H1(c1) × Hn−1(c2) in Hn+1
1 (c) has two distinct principal curvatures
±√c− c1 with multiplicity 1 and ∓√c− c2 with multiplicity n− 1. The square
of the norm of the second fundamental form satisfies (1.3).
U-H. Ki et al. [9] proved that
Theorem 1.4 ([9]). Let M be a complete space-like hypersurface with con-
stant mean curvature in an (n + 1)-dimensional Lorentzian space form Mn+1
1 (c).
If one of the following properties holds
(1) c ≤ 0,
(2) c > 0, n ≥ 3 and n2H2 ≥ 4(n− 1)c,
(3) c > 0, n = 2 and H2 > c,
then
S ≤ −nc +
n3H2
2(n− 1)
+
n(n− 2)
2(n− 1)
√
n2H4 − 4(n− 1)cH2, (1.2)
where S denotes the square of the norm of the second fundamental form of M .
As an application to Theorem 1.4, the authors of [9] gave a characterization
of hyperbolic cylinders of a Lorentzian space form Mn+1
1 (c) as follows:
Theorem 1.5 ([9]). The hyperbolic cylinders are the only complete space-
like hypersurfaces of Mn+1
1 (c) with non-zero mean curvature H and such that the
square of the norm of the second fundamental form satisfies
S = −nc +
n3H2
2(n− 1)
+
n(n− 2)
2(n− 1)
√
n2H4 − 4(n− 1)cH2. (1.3)
About the same time, R. Aiyama [13] obtained a characterization of hyper-
bolic cylinders of a Lorentzian 3-space form M3
1 (c) as follows:
Theorem 1.6 ([13]). The hyperbolic cylinders are the only complete space-
like surfaces in M3
1 (c) with non-zero constant mean curvature whose principal
curvatures λ and µ satisfy inf(λ− µ)2 > 0.
It is natural for us to pose the following problem:
Problem 1.1. Are the hyperbolic cylinders the only complete space-like hyper-
surfaces in an (n+1)-dimensional Lorentzian space form Mn+1
1 (c) with non-zero
constant mean curvature and two distinct principal curvatures λ and µ satisfying
inf(λ− µ)2 > 0?
We should note that for c ≤ 0, L. Cao and G. Wei [5] posed the same problem
as above, but they did not solve it. So the problem is still open.
In this paper, we will give a characterization of the hyperbolic cylinders of
(n+1)-dimensional Lorentzian space form Mn+1
1 (c), which implies that the above
Journal of Mathematical Physics, Analysis, Geometry, 2012, v. 8, No. 1 81
Shichang Shu and Annie Yi Han
Problem 1.1 can be solved affirmatively for c ≤ 0. For c > 0, we should note that
the condition H2 ≥ c is necessary. We state our result as follows:
Main Theorem. Let M be an n-dimensional (n ≥ 3) complete space-like
hypersurface in an (n+1)-dimensional Lorentzian space form Mn+1
1 (c) with non-
zero constant mean curvature and two distinct principal curvatures λ and µ of
multiplicities n− 1 and 1 and λ < µ. Then:
(1) For c = 0, if inf(λ− µ)2 > 0, then M is the hyperbolic cylinder H1(c1)×
Rn−1, where c1 < 0;
(2) For c < 0, if inf(λ− µ)2 > 0, then M is the hyperbolic cylinder H1(c1)×
Hn−1(c2), where 1
c1
+ 1
c2
= 1
c , c1 < 0, c2 < 0;
(3) For c > 0, if inf(λ−µ)2 > 0 and H2 ≥ c, then M is the hyperbolic cylinder
H1(c1)× Sn−1(c2), where 1
c1
+ 1
c2
= 1
c , c1 < 0, c2 > 0.
R e m a r k 1.1. For the case n = 2, this Main Theorem was proved by R.
Aiyama [13](see Theorem 1.6).
2. Preliminaries
Let M be an n-dimensional space-like hypersurface in an (n+1)-dimensional
Lorentzian space form Mn+1
1 (c). We choose a local field of the semi-Riemannian
orthonormal frames {e1, . . . , en+1} in Mn+1
1 (c) such that at each point of M ,
{e1, . . . , en} span the tangent space of M and form an othonormal frame there.
We use the following convention on the range of indices:
1 ≤ A,B, C, . . . ,≤ n + 1; 1 ≤ i, j, k, . . . ,≤ n.
Let {ω1, . . . , ωn+1} be the dual frame field so that the semi-Riemannian metric of
Mn+1
1 (c) is given by ds̄2 =
∑
i
ω2
i −ω2
n+1 =
∑
A
εAω2
A, where εi = 1 and εn+1 = −1.
The structure equations of Mn+1
1 (c) are given by
dωA +
∑
B
εBωAB ∧ ωB = 0, ωAB + ωBA = 0, (2.1)
dωAB +
∑
C
εCωAC ∧ ωCB = ΩAB, (2.2)
where
ΩAB = −1
2
∑
C,D
KABCDωC ∧ ωD, (2.3)
KABCD = εAεBc(δACδBD − δADδBC). (2.4)
If we restrict these forms to M , we have
ωn+1 = 0. (2.5)
82 Journal of Mathematical Physics, Analysis, Geometry, 2012, v. 8, No. 1
Characterization of Hyperbolic Cylinders in a Lorentzian Space Form
Cartan’s Lemma implies that
ωn+1i =
∑
j
hijωj , hij = hji. (2.6)
The structure equations of M are
dωi +
∑
j
ωij ∧ ωj = 0, ωij + ωji = 0, (2.7)
dωij +
∑
k
ωik ∧ ωkj = −1
2
∑
k,l
Rijklωk ∧ ωl, (2.8)
Rijkl = c(δikδjl − δilδjk)− (hikhjl − hilhjk), (2.9)
where Rijkl are the components of the curvature tensor of M , and
h =
∑
i,j
hijωi ⊗ ωj (2.10)
is the second fundamental form of M .
From the above equation, we have
n(n− 1)(R− c) = S − n2H2, (2.11)
where n(n − 1)R is the scalar curvature of M, H is the mean curvature, and
S =
∑
i,j
h2
ij is the square of the norm of the second fundamental form of M .
The Codazzi equations are
hijk = hikj , (2.12)
where the covariant derivative of hij is defined by
∑
k
hijkωk = dhij −
∑
m
hmjωmi −
∑
m
himωmj . (2.13)
The second covariant derivative of hij is defined by
∑
l
hijklωl = dhijk −
∑
m
hmjkωmi −
∑
m
himkωmj −
∑
m
hijmωmk. (2.14)
Then we have the following Ricci identities
hijkl − hijlk =
∑
m
hmjRmikl +
∑
m
himRmjkl. (2.15)
Journal of Mathematical Physics, Analysis, Geometry, 2012, v. 8, No. 1 83
Shichang Shu and Annie Yi Han
In a neighbourhood of a point x of M , we may choose orthonormal frame field
{e1, . . . , en} such that hij = λiδij at x. We introduce the operator φ given by
〈φX, Y 〉 = 〈hX, Y 〉 −H〈X, Y 〉. (2.16)
Putting φ =
∑
i,j
φijωi ⊗ ωj , where φij = hij − Hδij , we can easily see that φ is
traceless, that the basis {e1, . . . , en} also diagonalizes φ at x with eigenvalues
µi = λi −H, and that
|φ|2 =
∑
i
µ2
i =
1
2n
∑
i,j
(λi − λj)2 = S − nH2.
Therefore, we know that |φ|2 ≡ 0 if and only if M is totally umbilical. We shall
prove the following Lemma:
Lemma 2.1. Let M be an n-dimensional space-like hypersurface in an (n+1)-
dimensional Lorentzian space form Mn+1
1 (c) with constant mean curvature and
two distinct principal curvatures λ and µ of multiplicities n−1 and 1 and λ < µ.
Then
1
2
∆|φ|2 = |∇φ|2 + |φ|2{|φ|2 − n(n− 2)H√
n(n− 1)
|φ|+ n(c−H2)}. (2.17)
P r o o f. We firstly need the following result due to [14] and [15] : Let
µ1, µ2, . . . , µn be real numbers such that
∑
i
µi = 0 and
∑
i
µ2
i = β2, where β =
const ≥ 0, then
− n− 2√
n(n− 1)
β3 ≤
∑
i
µ3
i ≤
n− 2√
n(n− 1)
β3, (2.18)
and equality holds in the right-hand(left-hand) side if and only if (n − 1) of the
µ′is are non-positive and equal ((n− 1) of the µ′is are non-negative and equal).
Now we put µi = λi − H, then
∑
i
µi = 0 and
∑
i
µ2
i = |φ|2. Since M has
two distinct principal curvatures λ and µ of multiplicities n − 1 and 1, without
loss of generality, we may assume λ1 = · · · = λn−1 = λ, µ = λn, where λi for
i = 1, 2, . . . , n are the principal curvatures of M . Therefore, we know that
(n− 1)λ + µ = nH, S = (n− 1)λ2 + µ2. (2.19)
Since we assume that λ < µ, from (2.19), we have n(λ − H) = λ − µ < 0.
Therefore, we know that µ1 = · · · = µn−1 = λ − H < 0. We infer that the
equality holds in the right-hand side of (2.18), that is,
∑
i
µ3
i =
n− 2√
n(n− 1)
|φ|3. (2.20)
84 Journal of Mathematical Physics, Analysis, Geometry, 2012, v. 8, No. 1
Characterization of Hyperbolic Cylinders in a Lorentzian Space Form
From [8] or [10], we have the well-known Simons’ formula of Lorentzian version
as follows
1
2
∆|φ|2 = |∇φ|2 + (|φ|2)2 − nHtrφ3 + n(c−H2)|φ|2. (2.21)
From (2.20) and (2.21), we see that Lemma 2.1 is true.
The following generalized maximum principle will be important in the sequel.
Proposition 2.1 ([16, 17]). Let M be a complete Riemannian manifold with
Ricci curvature bounded from below and f a C2-function which is bounded from
below on M . Then there is a point sequence xk in M such that
lim
k→∞
f(xk) = inf(f), lim
k→∞
|∇f(xk)| = 0, lim
k→∞
inf ∆f(xk) ≥ 0.
Now we state a proposition which can be proved by making use of the similar
method due to Otsuki [18].
Proposition 2.2. Let M be a hypersurface in an (n + 1)-dimensional
Lorentzian space form Mn+1
1 (c) such that the multiplicities of the principal cur-
vatures are constant. Then the distribution of the space of the principal vectors
corresponding to each principal curvature is completely integrable. In particular,
if the multiplicity of a principal curvature is greater than 1, then this principal
curvature is constant on each integral submanifold of the corresponding distribu-
tion of the space of the principal vectors.
3. Proof of Main Theorem
We denote the integral submanifold through x ∈ Mn corresponding to λ by
Mn−1
1 (x). Putting
dλ =
n∑
k=1
λ,k ωk, dµ =
n∑
k=1
µ,k ωk. (3.1)
From Proposition 2.2, we have
λ,1 = λ,2 = · · · = λ,n−1 = 0 on Mn−1
1 (x). (3.2)
From (2.19), we have
dµ = −(n− 1)dλ. (3.3)
Hence, we also have
µ,1 = µ,2 = · · · = µ,n−1 = 0 on Mn−1
1 (x). (3.4)
Journal of Mathematical Physics, Analysis, Geometry, 2012, v. 8, No. 1 85
Shichang Shu and Annie Yi Han
From (2.13), we have
∑
k
hijkωk = dλiδij + (λj − λi)ωij . (3.5)
We infer that
hijk = 0, for any k, if i 6= j, 1 ≤ i ≤ n− 1 and 1 ≤ j ≤ n− 1. (3.6)
From (3.1), (3.2) and (3.5), we have for 1 ≤ j ≤ n− 1,
dλ = dλj =
n∑
k=1
hjjkωk
=
n−1∑
k=1
hjjkωk + hjjnωn = λ,n ωn.
(3.7)
Therefore, we have for 1 ≤ j ≤ n− 1,
hjjk = 0, 1 ≤ k ≤ n− 1, and hjjn = λ,n . (3.8)
From (3.1), (3.4) and (3.5), we have
dµ = dλn =
n∑
k=1
hnnkωk
=
n−1∑
k=1
hnnkωk + hnnnωn =
n∑
i=1
µ,i ωi = µ,n ωn.
(3.9)
Hence, we obtain
hnnk = 0, 1 ≤ k ≤ n− 1, and hnnn = µ,n . (3.10)
Now we prove the following Lemma:
Lemma 3.1. Let M be an n-dimensional space-like hypersurface in an (n+1)-
dimensional Lorentzian space form Mn+1
1 (c) with a constant mean curvature and
two distinct principal curvatures λ and µ of multiplicities n− 1 and 1. Then
|∇|φ|2|2 =
4n|φ|2
n + 2
|∇φ|2, (3.11)
where φ is defined by (2.16).
P r o o f. From (2.19), we have
|φ|2 = S − nH2 = n(n− 1)λ2 − 2n(n− 1)λH + n(n− 1)H2 (3.12)
= n(n− 1)(λ−H)2.
86 Journal of Mathematical Physics, Analysis, Geometry, 2012, v. 8, No. 1
Characterization of Hyperbolic Cylinders in a Lorentzian Space Form
Hence, from (3.2) we obtain
|∇|φ|2|2 =
∑
k
(|φ|2,k)2 =
∑
k
[2n(n− 1)(λ−H)λ,k]2 (3.13)
= 4n2(n− 1)2(λ−H)2(λ,n)2.
Since φij = hij −Hδij , from (3.3), (3.6), (3.8) and (3.10), we have
|∇φ|2 = |∇h|2 =
∑
i,j,k
h2
ijk =
n−1∑
i,j,k=1
h2
ijk + 3
n−1∑
i,j=1
h2
ijn + 3
n−1∑
i=1
h2
inn + h2
nnn (3.14)
= 3
n−1∑
i=1
h2
iin + h2
nnn = 3(n− 1)(λ,n)2 + (µ,n)2
= 3(n− 1)(λ,n)2 + (n− 1)2(λ,n)2 = (n− 1)(n + 2)(λ,n)2.
From (3.12), (3.13) and (3.14), we have
|∇|φ|2|2 = 4n2(n− 1)2(λ−H)2
|∇φ|2
(n− 1)(n + 2)
=
4n2(n− 1)(λ−H)2
n + 2
|∇φ|2 =
4n|φ|2
n + 2
|∇φ|2.
So, the proof of Lemma 3.1 is completed.
P r o o f of Main Theorem. Since we assume that inf(λ− µ)2 > 0, we have
(λ−µ)2 > 0. Putting (λ−µ)2 = κ > 0, we have [n(λ−H)]2 = (λ−µ)2 = κ > 0.
Therefore, we know that
|φ|2 = n(n− 1)(λ−H)2 =
n− 1
n
κ > 0,
that is, M is not umbilical. From Lemma 2.1 and Lemma 3.1, we have
1
2
∆|φ|2 =
n + 2
4n|φ|2 |∇|φ|
2|2 + |φ|2{|φ|2 − n(n− 2)H√
n(n− 1)
|φ|+ n(c−H2)}. (3.15)
Since the Ricci curvature Rii ≥ (n− 1)c− n2H2
4 and |φ|2 = n(n− 1)(λ−H)2 =
n−1
n κ > 0 are bounded from below, from Proposition 2.1, we have that there is a
point sequence xk in M such that
lim
k→∞
|φ|2(xk) = inf(|φ|2), lim
k→∞
|∇|φ|2(xk)| = 0, lim
k→∞
inf ∆|φ|2(xk) ≥ 0.
By (3.15), we have
inf |φ|2{inf |φ|2 − n(n− 2)H√
n(n− 1)
inf |φ|+ n(c−H2)} ≥ 0.
Journal of Mathematical Physics, Analysis, Geometry, 2012, v. 8, No. 1 87
Shichang Shu and Annie Yi Han
Since inf |φ|2 > 0, we have
inf |φ|2 − n(n− 2)H√
n(n− 1)
inf |φ|+ n(c−H2) ≥ 0. (3.16)
Since for c > 0, H2 ≥ c implies n2H2 ≥ 4(n−1)c, we know that the discriminant
of (3.16) is non-negative for all c. From (3.16), we have
inf |φ| ≤ 1
2
√
n
n− 1
[(n− 2)H −
√
n2H2 − 4(n− 1)c], (3.17)
or
inf |φ| ≥ 1
2
√
n
n− 1
[(n− 2)H +
√
n2H2 − 4(n− 1)c]. (3.18)
Assume that (3.17) holds, if c ≤ 0, we have inf |φ| ≤ 1
2
√
n
n−1 [(n−2)H−nH] < 0,
which contradicts inf |φ|2 > 0; if c > 0, since we assume that H2 ≥ c, we have
inf |φ| ≤ 1
2
√
n
n−1 [(n−2)H−
√
n2H2 − 4(n− 1)c] ≤ 0, this is also in contradiction
to inf |φ|2 > 0. Therefore, we know that (3.18) holds, we have
|φ|2 ≥ 1
4
n
n− 1
[(n− 2)H +
√
n2H2 − 4(n− 1)c]2,
and this is equivalent to
S ≥ −nc +
n3H2
2(n− 1)
+
n(n− 2)
2(n− 1)
√
n2H4 − 4(n− 1)cH2.
From Theorem 1.4, we have
S = −nc +
n3H2
2(n− 1)
+
n(n− 2)
2(n− 1)
√
n2H4 − 4(n− 1)cH2.
By Theorem 1.5, we see that Main Theorem is true.
Acknowledgment. The authors would like to thank the referee for his/her
many valuable suggestions and comments made that significantly improved the
paper.
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