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Cones in the Euclidean space with vanishing scalar curvature

Abstracts

Given a hypersurface M on a unit sphere of the Euclidean space, we define the cone based on M as the set of half-lines issuing from the origin and passing through M. By assuming that the scalar curvature of the cone vanishes, we obtain conditions under which bounded domains of such cone are stable or unstable.

stability; r-curvature; cone; scalar curvature


Dada uma hipersuperfície M de uma esfera unitária do espaço euclidiano, definimos o cone sobre M como o conjunto das semi-retas que saem da origem e passam por M. Admitindo que a curvatura escalar de um dado cone é nula, estabelecemos condições para que os seus domínios limitados sejam estáveis ou instáveis.

estabilidade; r-curvatura; cone; curvatura escalar


MATHEMATICAL SCIENCES

Cones in the Euclidean space with vanishing scalar curvature

João Lucas M. BarbosaI; Manfredo do CarmoII, * * Member Academia Brasileira de Ciências

IUniversidade Federal do Ceará (UFC), Departamento de Matemática, Campus do Pici, 60455-760 Fortaleza, Ce, Brasil

IIInstituto Nacional de Matemática Pura e Aplicada (IMPA), Estrada Dona Castorina 110, 22460-320 Rio de Janeiro, RJ, Brasil

Correspondence Correspondence to João Lucas M. Barbosa E-mail: jlucas@secrel.com.br

ABSTRACT

Given a hypersurface M on a unit sphere of the Euclidean space, we define the cone based on M as the set of half-lines issuing from the origin and passing through M. By assuming that the scalar curvature of the cone vanishes, we obtain conditions under which bounded domains of such cone are stable or unstable.

Key words: stability, r-curvature, cone, scalar curvature.

RESUMO

Dada uma hipersuperfície M de uma esfera unitária do espaço euclidiano, definimos o cone sobre M como o conjunto das semi-retas que saem da origem e passam por M. Admitindo que a curvatura escalar de um dado cone é nula, estabelecemos condições para que os seus domínios limitados sejam estáveis ou instáveis.

Palavras-chave: estabilidade, r-curvatura, cone, curvatura escalar.

1 INTRODUCTION

A natural generalization of minimal hypersurfaces in Euclidean spaces was introduced in (Reilly 1973). Reilly considered the elementary symmetric functions Sr, r = 0,1,¼,n, of the principal curvatures k1, ¼, kn of an orientable hypersurface x : Mn ® Rn+1 given by

Here, , ¼, are the eigenvalues of A = –dg, where g : Mn ® Sn(1) is the Gauss map of the hypersurface. Reilly showed that orientable hypersurfaces with Sr+1 = 0 are critical points of the functional

for variations of M with compact support. Thus, such hypersurfaces generalize the fact that minimal hypersurfaces are critical points of the area functional A0 = òMS0dM for compactly supported variations.

A breakthrough in the study of these hypersurfaces occurred in the last five years of last century: in (Hounie and Leite 1995) and (Hounie and Leite 1999) conditions for the linearization of the partial differential equation Sr+1 = 0 to be an elliptic equation were found. This linearization involves a second order differential operator Lr (see the definition of Lr in Section 2) and the Hounie-Leite conditions read as follows:

Lr is elliptic Û rank(A) > r + 1 Û Sr+2¹ 0 everywhere.

In this paper, we will be interested in the case S2 = 0. For this situation, since rank(A) cannot be two, the ellipticity condition is equivalent to rank (A) > 3.

In (Alencar et al. 2003) a general notion of stability was introduced for bounded domains of hypersurfaces of Euclidean spaces with Sr+1 = 0. In the case we are interested, namely S2 = 0, it can be shown that if we assume that L1 is elliptic, an orientation can be chosen so that a bounded domain D Ì M is stable if

> 0 for all variations with support in (the open set) D .

In what follows, we denote by Br(0) the ball of radius r centered at the origin 0 of Rn+1. Let Mn–1 be a smooth hypersurface of the sphere Sn(1). A cone (M) in Rn+1 is the union of half-lines starting at 0 and passing through the points of M. It is clear that (M) Ç Sn(1) = M. It is easy to show that (M) – {0} is a smooth n-dimensional hypersurface of Rn+1. The manifold (M) is referred to as the cone based on Mn–1. The part of the cone contained in the closure of the ring B1(0) \ Be(0), 0 < e < 1, is called a truncated cone and is denoted by (M)e.

In this note we present the following two theorems which provide a nice description of the stability of truncated cones in Rn+1 based on compact, orientable hypersurfaces of Sn(1), with S2 = 0 and S3¹ 0 everywhere.

THEOREM 1. Let Mn–1, n > 4, be an orientable, compact, hypersurface of Sn(1) with S2 = 0 and S3¹ 0 everywhere. Then, if n < 7, there exists an e > 0 so that the truncated cone (M)eis not stable.

THEOREM 2. For n > 8, there exist compact, orientable hypersurfaces Mn–1 of the sphere Sn(1), with S2 = 0 and S3¹ 0 everywhere, so that, for all e > 0, (M)e is stable.

Although Theorems 1 and 2 are interesting on their own right, a further motivation to prove these theorems is that, for the minimal case, they provide the geometric basis to prove the generalized Bernstein theorem, namely, that a complete minimal graph y = f(x1,¼, xn–1) in Rn, n < 8, is a linear function (See (Simons 1968), Theorems 6.1.1, 6.1.2, 6.2.1, 6.2.2).

For elliptic graphs in Rn with vanishing scalar curvature the question appears in a natural way. Of course, since we want to consider graphs with S2 = 0 and S3 never zero, we must start with n > 4, and the solution cannot be a hyperplane. Thus the question is whether there exists an elliptic graph in Rn, n > 4, with vanishing scalar curvature.

2 PRELIMINARIES

For notational reasons, it will be convenient to denote the hypersurface of the Introduction by x : ® Rn+1. We first need to consider the Newton Transformations Pr, that are inductively given by

and then define the differential operator Lr by

It turns out that Lr is self-adjoint and that Lr f = div(Prgrad f).

The second variation formula for the variational problem of the functional

1 is, up to a positive constant, given by

for test functions f of compact support in .

Consider now a compact orientable (n – 1)-dimensional manifold M immersed as a hypersurface of the unit sphere Sn(1) of the Euclidean space Rn+1. The cone (M) based on M is described by

Of course, the geometry of (M) is closely related to the one of M and it is simple to compute the second fundamental form of (M) in terms of the second fundamental form A of M. In fact, one finds

From this relation on the second fundamental forms it follows that

PROPOSITION 1. If

r represents the elementary symmetric function of order r of (M) and r its Newton transformations, then:

a)

r = (1 /tr) Sr,

b)

r = 0 if and only if Sr = 0 ,

c) || = (1/t) |A| ,

d) r = (1/tr) .

PROOF. The proof is direct except for the last item. But this can be done using finite induction and the definition of r.

Let F : (M) ® R be a C2 function. For each t > 0, define t : M ® R by t(m) = F(m, t).

Proposition 2. With the above notation we have:

PROOF. The proof of this Lemma follows the same lines used to find the expression of the Laplacian in polar coordinates and using the previous proposition.

3 SKETCH OF PROOF OF THEOREM 1

First of all let us observe that since S2º 0 then (S1)2 = |A|2> 0. Hence, at a point where S1 = 0 we would have that all the entries of the matrix A are zero and so S3 = 0 what is forbidden by our hypothesis. Therefore, we will have (S1)2 > 0 everywhere.

According to Proposition 1, our hypotheses then imply that, for the cone (M), we have 2º 0 and 1 and 3 never zero.

It was proved in (Hounie and Leite 1999) that, for a hypersurface of Rn+1 with r º 0, 2 < r < n, the operator r–1 is elliptic if and only if r+1 is never zero. Then we conclude that L1 and 1 are elliptic.

To prove the theorem, we are going to show the existence of a truncated cone (M) for which the second variation formula attains negative values. Hence, from now on we are going to work on a truncated cone, with test functions f that have a support contained in the interior of the truncated cone. As we did before, for each test function f : (M)® R and each fixed t we define t : M ® R by t(m) = f(m, t). From Proposition 2 we have that

The volume element of (M) is easily seen to be

Hence, using (3), (5) and the expression of the volume, the second variation formula on f becomes

Since S1 > 0, then tn–4S1dt Ù dM is a volume element in (M), in particular in (M) . We will represent it by dS. In fact, dS is a product of two measures. The first one on the real line: dx = tn–4dt; the second, on M, given by dm = S1dM. So, dS = dx Ù dm. We can then rewrite the second variation formula on f as:

Define, now, the following two operators:

Observe that we are considering the space C¥(M) with the inner product:

and C¥[, 1] with the inner product:

Since L1 is elliptic and M is compact then L1, and so 1, is strongly elliptic. The same is true for the operator 2. Let l1< l2¥ be the eigenvalues of 1 and d1 < d2 < ¼ ¥ be the eigenvalues of 2. Using orthonormal bases of eigenfunctions for theses operators one deduces the following Lemma:

LEMMA 1. For any test function f we have

There exists a test function f such that I(f) < 0 if and only if l1 + d1 < 0.

The operator

2 is well known. In fact it has been used in (Simmons 1968) to prove his celebrated theorem. The following lemma contains all the information we need about this operator:

LEMMA 2. The operator

2 has eigenvalues

where 1 <k < ¥.

We will also need the following lemma whose proof uses Lemmas (3.7) and (4.1) in (Alencar et al. 1993).

LEMMA 3. Let Mn–1be a compact, orientable, immersed hypersurface of Sn(1) with S2º 0 e S3 never zero. Suppose n > 4. The first eigenvalue of the operator

1 in M satisfy: l1< -(n – 2).

Finally, we observe that the lemma below completes the proof of Theorem 1.

LEMMA 4. Let Mn–1be a compact, orientable, immersed hypersurface of Sn(1) with S2º 0, S3 never zero and n > 4. If n < 7 then there exists

> 0 such that the truncated cone M is not stable.

PROOF OF THE LEMMA: From Lemmas 2 and 3 we have

It is trivial to verify that the sum of the first two terms of the right hand side of this inequality is a quadratic polynomial, with positive second order term, whose roots are approximately 2.2 and 7.8 . Hence it is strictly negative for values of n Î {4,5,6,7 }, in fact, it is less than or equal to -1. Hence,

Choosing sufficiently small we can guarantee that the right hand side is negative. Now, by Lemma 1, we see that M is not stable. This proves Lemma 4 and completes the proof of the Theorem 1.

4 EXISTENCE OF STABLE CONES

In this section we sketch the proof o Theorem 2.

The following example has been considered by various people in different contexts (see e.g. (Chern 1968) and (Alencar et al. 2002). Consider Rp+2 = Rr+1Å Rs+1, r + s = p. Write down the vectors of Rp+2 as x1 + x2, x1Î Rr+1, x2Î Rs+1. When x1 describes Sr(1) Ì Rr+1 and x2 describes Ss(1) Ì Rs+1, by taking positive numbers a1 and a2 with + = 1, we have that

x = a1x1 + a2x2

describes a submanifold M of dimension p = r + s of the sphere Sp+1(1) Ì Rp+2. The manifold M is diffeomorphic to Sr(1) × Ss(1) and so is compact and orientable. It can be shown that a1 and a2 can be chosen so that S2 = 0 and S3¹ 0. We will show that, in this case, the truncated cone (M)e is stable as a hypersurface of Rr+s+1 when r + s + 1 > 8.

It can be shown that the L1 operator on M is given by

where Dr and Ds denote the Laplacian operators in the Euclidean spheres Sr(a1) and Ss(a2), respectively. Since the first nonzero eigenvalue of the Laplace operator on a sphere Sk(b) is known to be k/b2, the first nonzero eigenvalue of L1 will be

It will then follow that the first eigenvalue of the operator

will be given by

where the last equality comes from a long but straightforward computation.

Therefore, using Lemma 2, the above value for l1, and the fact that, in our case, n = p + 1, we obtain

For n > 8, the sum of the first two terms becomes > 1/4. Thus, for any choice of e, l1 + d1 > 0. Together with Lemma 1, this completes the proof of Theorem 2.

ACKNOWLEDGMENTS

Both authors were partially supported by PRONEX and CNPq.

Manuscript received on April 22, 2004; accepted for publication on May 12, 2004.

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  • Correspondence to
    João Lucas M. Barbosa
    E-mail:
  • *
    Member Academia Brasileira de Ciências
  • Publication Dates

    • Publication in this collection
      22 Nov 2004
    • Date of issue
      Dec 2004

    History

    • Received
      22 Apr 2004
    • Accepted
      12 May 2004
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