On complete spacelike hypersurfaces with constant scalar curvature in the de Sitter space

Abstract

Let Mn be a complete spacelike hypersurface with constant normalized scalar curvature R in the de Sitter Space S1n + 1. Let H the mean curvature and suppose that <img ALIGN="BOTTOM" BORDER="0" SRC="http:/img/fbpe/aabc/v72n4/0045img1.gif" ALT="$ \overline{R}$"> = (R - 1) > 0 and <img ALIGN="BOTTOM" BORDER="0" SRC="http:/img/fbpe/aabc/v72n4/0045img1.gif" ALT="$ \overline{R}$"> <= sup H² <= C<img ALIGN="BOTTOM" BORDER="0" SRC="http:/img/fbpe/aabc/v72n4/0045img3.gif" ALT="$\scriptstyle \overline{R}$">, where C<img ALIGN="BOTTOM" BORDER="0" SRC="http:/img/fbpe/aabc/v72n4/0045img3.gif" ALT="$\scriptstyle \overline{R}$"> is a constant depending only on R and n. It is proved that either sup H² = <img ALIGN="BOTTOM" BORDER="0" SRC="http:/img/fbpe/aabc/v72n4/0045img1.gif" ALT="$ \overline{R}$"> and Mn is totally umbilical, or sup H² = C<img ALIGN="BOTTOM" BORDER="0" SRC="http:/img/fbpe/aabc/v72n4/0045img3.gif" ALT="$\scriptstyle \overline{R}$"> and Mn is the hyperbolic cylinder H¹(1 - coth²r) x Sn - 1 (1 - tanh²r).

hyperbolic cylinder; spacelike hypersurfaces; de Sitter space

On Complete Spacelike Hypersurfaces with Constant Scalar Curvature in the De Sitter Space

ALDIR BRASIL JR and A. GERVASIO COLARES

Universidade Federal do Ceará, Departamento de Matemática,

Campus do Pici - 60455-760 Fortaleza, CE, Brazil

Manuscript received on April 2, 2000; accepted for publication on May 10, 2000;

presented by MANFREDO DO CARMO

ABSTRACT

MⁿRS₁^{n + 1}HR H²CCRn H²Mⁿ H²CMⁿH¹²rxS^{n- 12}r

Key words: hyperbolic cylinder, spacelike hypersurfaces, de Sitter space.

1. INTRODUCTION

The study of spacelike hypersurfaces with constant scalar curvature in the de Sitter space S₁^{n + 1}is related to an analogue of Goddard's conjecture for the second elementary symmetric polynomial in the principal curvatures; more precisely: "Let Mⁿ be a complete spacelike hypersurface with constant scalar curvature immersed in de Sitter space S₁^{n + 1}. Then Mⁿ is totally umbilical ''.

S. Montiel (Montiel 1996) described the hyperbolic cylinders H¹(1 - coth²r) x S^{n - 1}(1 - tanh²r), n 3, in S₁^{n + 1} with constant mean curvature H² = and normalized constant scalar curvature R = 1 + (2 + (n - 2)tanh²r)).

In (Zheng 1995), (Zheng 1996) and (Cheng & Ishikawa 1998) partial results were obtained. Recently, Haizhong Li (Li 1997)(and also S. Montiel in a more general spacetime (Montiel 1999)) obtained the following result: "Let Mⁿ be a compact spacelike hypersurface immersed into the de Sitter space S₁^{n + 1} with normalized constant scalar curvature R satisfying R 0. Then M is totally umbilical".

In this note we announce the following

THEOREM 1.1. Let Mⁿ be an n-dimensional complete (n 3) spacelike hypersurface immersed into the de Sitter space S₁^{n + 1} with constant scalar curvature R such that = R - 1 > 0 and supose that supH²

C where

C = (n - 1)² + 2(n - 1) + .

Then

(i)
• sup H² = and M is totally umbilic or
(ii)
• sup H² = C and M is isometric to H¹(1 - coth²r) x S^{n - 1}(1 - tanh²r).

2. PRELIMINARIES

Let R₁^{n + 2} be the real vector space R^{n + 2} endowed with the Lorentzian metric , given by v, w = - v₀w₀ + v₁w₁ +...+ v_{n + 1}w_{n + 1} that is, R₁^{n + 2} = L^{n + 2} is the Lorentz-Minkowski (n + 2)-dimensional space. We define the de Sitter space as the following hyperquadric of R₁^{n + 2} : S₁^{n + 1} = {x

R₁^{n + 2};| x|² = 1}. The induced metric , makes S₁^{n + 1} into a Lorentz manifold with constant sectional curvature 1. Let Mⁿ be a n-dimensional orientable manifold, complete and let f : Mⁿ S₁^{n + 1}

L^{n + 2} be a spacelike immersion of Mⁿ into the de Sitter S₁^{n + 1}. Choose a unit normal along f and denote by A : T_pM T_pM the linear map of the tangent space T_pM at the point p M, associated to the second fundamental form of f along ,

AX, Y = -

Y,,

where X and Y are tangent vector fields on M and is the connection on S₁^{n + 1}. Let {e₁,..., e_n} be an orthonormal basis which diagonalizes A with eigenvalues k_i of T_pM, i.e., Ae_i = k_ie_i, i = 1,..., n. We will denote by H =


k_i the mean curvature of f and | A|² = k_i². In our case it is convenient to define a linear map : T_pM T_pM by

X, Y = AX, Y - H

X, Y.

It is easily checked that trace() = 0 and that

||² = (k_i - k_j)²,

so that ||² = 0 if and only if Mⁿ is totally umbilical. Let = k_i - H be the eigenvalues of ; then = 0, and

|

(2.1)

The standard examples of spacelike umbilical hypersurfaces with constant mean curvature in the de Sitter space are given by

Mⁿ = {p S₁^{n + 1}| p, a = },

where a R₁^{n + 2}, | a|² = = 1, 0, - 1 and > . The corresponding mean curvature H of such surfaces satisfies

H² =

(Montiel 1988) and Mⁿ is isometric to a hyperbolic space, an Euclidean space or a sphere according to equal to 1, 0, - 1, respectively. On the other hand, hyperbolic cylinders are the hypersurfaces given by

Mⁿ = {p S₁^{n + 1}; - p_o² + p₁² + ... + p_k² = - sinh²r},

with r R and 1 k n.

Such hyperbolic cylinders have constant mean curvature

nH = [kcothr + (n - k)tanh r].

Thus, we have

H²




and the equality is attained for k = 1 and coth²r = (n - 1). Their normalized scalar curvature is R = 1 + (2 + (n - 2)tanh²r).

We point out that these examples have only two different constant principal curvatures at each point and one of them has multiplicity one. Moreover, they are isometric to the Riemannian product

H¹(1 - coth²r) x S^{n - 1}(1 - tanh²r).

The Gauss equation relates the scalar curvature, the mean curvature and the square of the norm of the second fundamental formula as follows:

n(n - 1)(R - 1) = n²H² - | A|². (2.2)

Let T = T_ij

be a symmetric tensor defined on Mⁿ, where

T_ij = nH - h_ij.

Following (Cheng & Yau 1977), we introduce the operator L₁ associated to T acting on C² functions f on Mⁿ by

(2.3)

Around a given point p M we choose an orthonormal frame field {e₁,..., e_n} with dual frame field {w₁,..., w_n} so that h_ij = k_i at p. We have the following computation by using (2.3) and Gauss equation (2.2) :

(2.4)

On the other hand, using Simons Formula (see Zheng 1995) we get

|

(2.5)

From (2.4) and (2.5), we have:



(2.6)

3. SKETCH OF THE PROOF OF THE THEOREM

Since R is constant, by (2.6) we obtain

L₁(nH) = A - n²H + (1 + k_ik_j)(k_i - k_j)² = A - n²H + nk_i² + nk_j² - k_jk_i + k_i³k_j + k_ik_j³ - k_i²k_j².

Making i = j, we then have

(3.1)

Using (2.1) in (3.1) we obtain



(3.2)

Then,



(3.3)

This yields



(3.4)

We need to estimate tr() in (3.4). First we recall an algebraic lemma (Okumura 1974) which asserts

(|

(3.5)

and the equality holds on the right hand side if and only if

= ... = = - || and = ||.

Using (3.5) in (3.4), we obtain



(3.6)

Since R is constant, by (Alencar et al. 1993)



(3.7)

By Gauss equation (2.1) we know that



(3.8)

Using (3.8) and (3.7) in (3.6) we obtain



(3.9)

where P_H is a polynomial given by

|) = (

(3.10)

By (3.8) we may write the above polynomial as



(3.11)

Therefore (3.9) becomes



(3.12)

Using the hypothesis that


supH²

C, one proves that
(

(3.13)

On the other hand,



(3.14)

where | H|_max is the maximum of the mean curvature H in M and C = mink_i is the minimum of the principal curvatures in M.

Now, we need the maximum principle at infinity for complete manifolds by Omori and Yau(Omori 1967):

"Let Mⁿ be an n-dimensional complete Riemannian manifold whose Ricci curvature is bounded from below. Let f be a C²-function bounded from below on Mⁿ. Then for each > 0 there exists a point p



M such that


(3.15)

The hypothesis


supH²

C together with Gauss equation implies that Ric_M (n - 1) - , so the Ricci curvature is bounded below. Thus we may apply Omori and Yau's result to the function

f = .

We have



(3.16)

and



(3.17)

Let {p_k}, k N, be a sequence of points in M given by (3.15) such that



(3.18)

Using (3.18) in the two equations (3.16) and (3.17) and the fact that

(nH)(p_k) = (nH)(p),

we obtain

< -

(3.19)

Hence

(

(3.20)

On the other hand, by (3.12) and (3.14), we have

(|

(3.21)

At points p_k of the sequence given in (3.18), this becomes



(3.22)

Making k and using (3.20) we have that the right hand side of (3.22) goes to zero, so by (3.15) either (sup| A|² - n) = 0 or P() = 0. But by (3.8) ||² = (| A|² - n) and so sup||² = (sup| A|² - n) = 0, then ||² = 0, proving that Mⁿ is totally umbilical.

If P() = 0, it can be proved that supH² = C and so the equality holds on the right hand side of (3.9) and we obtain

L₁(nsupH) = (sup| A|² - n)P().

One proves that the equality also holds in Okumura's lemma (3.5). After reenumeration, we finally have

k₁ = k₂ =...= k_{n - 1}, k₁


k_n, wherek₁ = tanhr andk_n = coth r.

Therefore, Mⁿ is isometric to H¹(1 - coth²r) x S^{n - 1}(1 - tanh²r), finishing the proof.

Correspondence to: A. Gervasio Colares

E-mail: [e-mail]

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Publication Dates

• Publication in this collection
  05 Jan 2001
• Date of issue
  Dec 2000

History

• Received
  02 Apr 2000
• Accepted
  10 May 2000