SPACE WITH CONSTANT HIGHER-ORDER
MEAN CURVATURE
KAIREN CAI AND HUIQUN XU
Received 26 March 2006; Accepted 26 March 2006
The authors apply the generalized Minkowski formula to set up a spherical theorem. It is shown that a compact connected hypersurface with positive constant higher-order mean curvatureHr for some fixedr, 1≤r≤n, immersed in the de Sitter spaceSn1+1must be a sphere.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction
The classical Liebmann theorem states that a connected compact surface with constant Gauss curvature or constant mean curvature inR3is a sphere. The natural generalizations of the Gauss curvature and mean curvature are therth mean curvatureHr,r=1,...,n, which are defined as therth elementary symmetric polynomial in the principal curva-tures ofM. Later many authors [1,4,5,7,8] have generalized Liebmann theorem to the cases of hypersurfaces with constant higher-order mean curvature in the Euclidian space, hyperbolic space, the sphere, and so on. A significant result due to Ros [8] is that a compact hypersurface with therth constant mean curvatureHr, for somer=1,...,n, embedded into the Euclidian space must be a sphere.
The purpose of this note is to prove a spherical theorem of the Liebmann type for the compact spacelike hypersurface immersed in the de Sitter space by setting up a general-ized Minkowski formula. The main result is the following.
Theorem 1.1. LetMbe a compact connected hypersurface immersed in the de Sitter space Sn+1
1 . If for some fixedr, 1≤r≤n, therth mean curvatureHris a positive constant onM, thenMis isometric to a sphere.
For the cases of the constant mean curvature and constant scalar curvature, that is, r=1, 2, the theorem was founded by Montiel [4] and Cheng and Ishikawa [1], respec-tively.
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 19545, Pages1–6
2. Preliminaries
LetRn+2
1 be the real vector spaceRn+2endowed with the Lorentzian metric·,·given by
x,y = −x0y0+ n+2 i=1
xiyi (2.1)
forx,y∈Rn+2. The de Sitter spaceSn+1
1 (c) can be defined as the following hyperquadratic:
Sn1+1(c)=
x∈Rn+2 1 | |x|2=
1 c,
1 c >0
. (2.2)
In this way, the de Sitter space inherits from·,·a metric which makes it an indefinite Riemannian manifold of constant sectional curvaturec. Ifx∈Sn1+1(c), we can put
TxSn1+1(c)=
v∈Rn+2
1 | v,x =0
. (2.3)
Letψ:M→Sn1+1be a connected spacelike hypersurface immersed in the de Sitter space with the sectional curvature 1. Following O’Neill [6], the unit normal vector fieldNfor ψcan be viewed as the Gauss map ofM:
N:M−→x∈Rn+2
1 | |x|2= −1
. (2.4)
Let Sr:Rn→R,r=1,...,n, be the normalizedrth elementary symmetric function in the variables y1,...,yn. For r=1,...,n, we denote byCr the connected component of the set{y∈Rn|S
r(y)>0} containing the vectory=(1,..., 1). Notice that every vec-tor (y1,...,yn) with all its components greater than zero lies in eachCr. We derive the following two lemmas, which will be needed for the proof of the theorem.
Lemma 2.1 [3]. (i) Ifr≥k, thenCr⊂Ck; (ii) fory∈Cr,
S1/r
r ≤S1r−/r−11≤ ··· ≤S12/2≤S1. (2.5)
Lemma 2.2 (Minkowski formula). Letψ:M→Sn+1
1 ⊂Rn1+2be a connected spacelike hy-persurface immersed in de Sitter spaceSn1+1. For therth mean curvatureHrofψ,r=0, 1,..., n−1,
M
Hrψ,a+Hr+1N,a dV=0, (2.6)
whereH0=1 anda∈Rn+1
1 is an arbitrary fixed vector andNis the unit normal vector ofM. Proof. The argument is based on the approach of geodesic parallel hypersurfaces in [5]. Let kr and ei,i=1,...,n, be the principal curvatures and the principal directions at a pointp∈M. Therth mean curvature ofψis defined by the identity
Pn(t)=1 +tk1 ···1 +tkn =1 +
n 1
H1t+···+
n n
for allt∈R. ThusH1=H is the mean curvature,H2=(n2H2−S)/n(n−1), whereSis the square length of the second fundamental form andHnis the Gauss-Kronecker curva-ture ofMimmersed inSn+1
1 . Let us consider a family of geodesic parallel hypersurfacesψt given by
ψt(p)=expψ(p)
−tN(p) =cosht·ψ(p) + sinht·N(p). (2.8)
Then the unit normal vector field ofψtwith|Nt|2= −1 can be written as
Nt(p)= −sinht·ψ(p)−cosht·N(p). (2.9)
Because we have
ψt∗
ei =cosht−kisinht ei , Nt∗
ei =
−sinht+kicosht ei ;
(2.10)
for the principal directions{ei},i=1,...,nand|t|< ε, the second fundamental form of ψtcan be expressed as
σt
ψt∗
ei ,ψt∗
ej
= −Nt∗
ei ,ψt∗
ej
=sinht−kicosht ei,ψt∗
ej
=sinht−kicosht cosht−kisinht
ψt∗
ei ,ψt∗
ej .
(2.11)
Then the mean curvatureH(t) ofψcan be expressed as
H(t)=1
n n
i=1
ki(t)=1 n
n
i=1
tanht−ki 1−kitanht
= 1
nPn(−tanht) n
i=1
tanht−ki j=i
1−kjtanht .
(2.12)
But
j=i
1−kjtanht =nPn(−tanht)−coshtsinht P
n(−tanht). (2.13)
Then we get
H(t)=tanht+ P
n(−tanht) nPn(−tanht).
(2.14)
By the way, we must point out that the formula (7) in [5] is incorrect because the second term in the right-hand side of the expression ofH(t) should bePn(tanht)/nPn(tanht). The volume elementdVtfor immersionψtcan be given by
dVt=cosht−k1sinht ···conht−knsinht dV
= −conhnt Pn(−tanht)dV,
wheredVis the volume element ofψ. It is an easy computation that
ψ,a+HN,a =0, (2.16)
whereN is a unit normal field ofψanda∈Rn+2
1 an arbitrary fixed vector (cf. [4, page 914]). Integrating both sides of (2.16) over the hypersurfaceMand applying Stoke’s the-orem, we get
M
ψ,a+H1N,a dV=0. (2.17)
Forψt,|t|< ε, we obtain
M
ψt,a
+H(t)Nt,a
dVt=0. (2.18)
Substituting (2.14) and (2.15) into (2.18), we get
M
ψt,a
+H(t)Nt,a dVt =1 ncosh n−1 t M
nPn(−tanht)−sinhtcoshtP
n(−tanht) ψ,a
−cosh2tPn(−tanht)N,a dV=0.
(2.19)
By using the expression
nPn(−tanht)−sinhtcoshtP
n(−tanht)
=n+ (n−1)
n 1
H1(−tanht) +···+n
n n−1
Hn(−tanht)n−1,
(2.20)
we obtain
M
nPn(−tanht)−sinhtcosht P
n(−tant) ψ,a −conh2tP
n(−tanht)N,a
dV
=n
r=1
(n−r−1)
n r−1
(−tanht)r−1,
M
Hr−1
ψt,a+HrNt,a dV=0.
(2.21)
The left-hand side in the equality is a polynomial in the variable tanht. Therefore, all its coefficients are null. This completes the proof ofLemma 2.2.
3. Proof ofTheorem 1.1
Here we work for the immersed hypersurfaces inSn+1
abstracting from (2.6), we obtain that
M
H1Hr−Hr+1 N,adV=0. (3.1)
We know from Newton inequality [2] thatHr−1Hr+1≤Hr2, where the equality implies thatk1= ··· =kn. Hence
Hr−1
H1Hr−Hr+1 ≥Hr
H1Hr−1−Hr . (3.2)
It derives fromLemma 2.1that
0≤H1/r
r ≤Hr−1/r−11≤ ··· ≤H21/2≤H1. (3.3)
Thus we conclude that
Hr−1
H1Hr−Hr+1 ≥Hr
H1Hr1−Hr ≥0, (3.4)
and ifr≥2, the equalities happen only at umbilical points ofM. We choose a constant vectorasuch that|a|2= −1 anda0≤ −1. Since the normal vectorNsatisfies|N|2= −1, we haveN,a ≥1 onM. It follows from (3.1) that
H1Hr−Hr+1=0. (3.5)
Thus,k1= ··· =kn,Mis totally umbilical, andMis isometric to a sphere. This ends the proof ofTheorem 1.1.
If there is a convex point onM, that is, a point at whichki>0, for alli=1,...,n, then the constantrth mean curvatureHris positive. By means of the same argument as that of Theorem 1.1, we derive the following.
Corollary 3.1. Let M be a compact connected hypersurface immersed in the de Sitter space
Sn+1
1 . If for some fixedr, 1≤r≤n, therth mean curvatureHr is constant, and there is a convex point onM, thenMis isometric to a sphere.
Acknowledgment
The project is supported by the Natural Science Foundation of Zhejiang Provence in China.
References
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[3] L. G˙arding, An inequality for hyperbolic polynomials, Journal of Mathematics and Mechanics 8 (1959), 957–965.
[5] S. Montiel and A. Ros, Compact hypersurfaces: the Alexandrov theorem for higher order mean curvatures, Differential Geometry. Proceedings Conference in Honor of Manfredo do Carmo, Pitman Monogr. Surveys Pure Appl. Math., vol. 52, Longman Scientific & Technical, Harlow, 1991, pp. 279–296.
[6] B. O’Neill, Semi-Riemannian Geometry. With Applications to Relativity, Pure and Applied Math-ematics, vol. 103, Academic Press, New York, 1983.
[7] R. C. Reilly, Applications of the Hessian operator in a Riemannian manifold, Indiana University Mathematics Journal 26 (1977), no. 3, 459–472.
[8] A. Ros, Compact hypersurfaces with constant higher order mean curvatures, Revista Matem´atica Iberoamericana 3 (1987), no. 3-4, 447–453.
Kairen Cai: Department of Mathematics, Hangzhou Teachers College, 222 Wen Yi Road, Hangzhou 310036, China
E-mail addresses:kcai53@hotmail.com; kairencai408@hzcnc.com
Huiqun Xu: Department of Mathematics, Hangzhou Teachers College, 222 Wen Yi Road, Hangzhou 310036, China
