Hypersurfaces with constant mean curvature and two principal curvatures in <img id="_x0000_i1025" src="../../../../img/revistas/aabc/v76n3/a03img01.gif" align=absbottom>n+1

MATHEMATICAL SCIENCES

Hypersurfaces with constant mean curvature and two principal curvatures in

ⁿ⁺¹

Luis J. Alías^I; Sebastião C. de Almeida^II; Aldir Brasil Jr.^III

^IDepartamento de Matemáticas, Universidad de Murcia E-30100 Espinardo - Murcia, Spain

^IIDEA-CAEN, Universidade Federal do Ceará, 60020-181 Fortaleza,CE, Brasil

^IIIDepartamento de Matemática, Universidade Federal do Ceará 60455-760 Fortaleza,CE, Brasil

^{Correspondence}

ABSTRACT

In this paper we consider compact oriented hypersurfaces M with constant mean curvature and two principal curvatures immersed in the Euclidean sphere. In the minimal case, Perdomo (Perdomo 2004) andWang (Wang 2003) obtained an integral inequality involving the square of the norm of the second fundamental form of M, where equality holds only if M is the Clifford torus. In this paper, using the traceless second fundamental form of M, we extend the above integral formula to hypersurfaces with constant mean curvature and give a new characterization of the H(r)-torus.

Key words: Hypersurfaces, constant mean curvature, Simons formula, H(r)-torus.

RESUMO

Neste trabalho tratamos das hipersuperfícies fechadas com curvatura média constante e duas curvaturas principais imersas na esfera euclidiana. No caso de hipersuperfícies mínimas, Perdomo (Perdomo 2004) e Wang (Wang 2003) obtiveram uma desigualdade integral envolvendo o quadrado da norma da segunda forma fundamental de M, onde ocorre a igualdade se e somente se M é o toro de Clifford. Neste trabalho, usando o segunda forma fundamental modificada com traço nulo de M, obtemos uma generalização da fórmula integral acima e uma nova caracterização dos H(r)-toros.

Palavras-chave: Hipersuperfícies, curvatura média constante, Fórmula de Simons, H(r)-toros.

1 INTRODUCTION

Let M be a compact minimal hypersurface of the (n + 1)-dimensional unit Euclidean sphere ⁿ⁺¹. As usual, let S be the square of the length of the second fundamental form A of M. If 0 < S < n, then Simons (Simons 1968) proved that either S = 0 or S = n. On the other hand, Do Carmo et al. 1970 and Lawson (Lawson 1969) proved, independently, that the Clifford tori are the only minimal hypersurfaces with S = n. The particular case n = 3 was studied by Peng and Terng. They proved in (Peng and Terng 1983) that if S > 3 is a constant function, then S > 6. In (Otsuki 1970) Otsuki proved that minimal hypersurfaces of ⁿ⁺¹ having distinct principal curvatures of multiplicities k and m = n - k > 2 are locally product of spheres of the type ^m(c₁) × ^n-m(c₂), and he constructed examples of compact minimal hypersurfaces in ⁿ⁺¹ with two distinct principal curvatures and one of them being simple. Recently, Hasanis and Vlachos (Hasanis and Vlachos 2000) proved that if M is a compact minimal hypersurface with two principal curvatures, one of them with multiplicity 1 and S > n, then S = n and M is a Clifford torus. Using a traceless tensor F = A - HI, the so called traceless second fundamental form, Alencar and do Carmo (Alencar and do Carmo 1994) proved that if Mⁿ is compact with constant mean curvature H and |F|²< B_H, where B_H is a constant that depends only on H and n, then either |F|² = 0 or |F|² = B_H. They also proved that the H(r)-tori ^n-1(r) × ¹() with r²< (n - 1)/n are the only hypersurfaces with constant mean curvature H and |F|² = B_H. These results do not characterize the other tori ^n-1(r) × ¹(), with r² > (n - 1)/n, nor ^n-k(r) × ^k() for 2 < k < n - 1. Recently, the third author jointly with Barbosa, Costa and Lazaro (Barbosa et al. 2003) obtained a generalization of the result of Hasanis and Vlachos without any additional hypothesis on the mean curvature. They obtained a characterization for the H(r)-tori ^n-1(r) × ¹() with r²> (n - 1)/n. More precisely:

THEOREM 1 (Barbosa et al. 2003). Let M be a compact oriented hypersurface immersed in the sphere ⁿ⁺¹, with two distinct principal curvatures l and m with multiplicities 1 and n - 1, respectively. Suppose in addition that n > 3 and |F|²> C_H, where

Then H is constant, |F|² = C_H and M is isometric to an H(r)-torus ^n-1(r) × ¹() with r²> (n - 1)/n.

Recently, Perdomo (Perdomo 2004) and Wang (Wang 2003) independently obtained the following integral formula for compact minimal hypersurfaces with two principal curvatures immersed in

ⁿ⁺¹.

THEOREM 2 (Perdomo 2004, Wang 2003). Let Mⁿ be a compact minimal hypersurface immersed in ⁿ⁺¹. If M has two principal curvatures everywhere, then

if l is a principal curvature with multiplicity n - 1.

A natural consequence of the integral formula above is that if M is a compact minimal hypersurface with two principal curvatures immersed in ⁿ⁺¹, then

with equality if and only if M is a Clifford hypersurface. In this paper we will extend the integral formula (1) for compact hypersurfaces with constant mean curvature and obtain a new characterization of the H(r)-torus ^n-1(r) × ¹(). Explicitly, we have the following result.

THEOREM 3. Let M be a compact oriented hypersurface immersed in the sphere ⁿ⁺¹with constant mean curvature H. Suppose in addition that M has two distinct principal curvatures l and m with multiplicities (n - 1) and 1, respectively. If F = A - HI is the traceless second fundamental form of M and P_H is the Alencar-do Carmo polynomial

then

where c = ±1 is the sign of the difference l - m.

The polynomial P_H was first introduced by Alencar and do Carmo in (Alencar and do Carmo 1994) in their study on hypersurfaces with constant mean curvature in the sphere. Actually, the sharp positive constant B_H found by them in that paper is given precisely as the square of the positive root of P_H (a constant that depends only on H and n). For that reason we will refer to P_H as the Alencar-do Carmo polynomial.

In the minimal case, we make H = 0 in the integral formula (2) and retrieve Perdomo' s and Wang' s integral formula (1). From the above result we obtain.

COROLLARY 4. Let M be a compact oriented hypersurface immersed in the sphere ⁿ⁺¹with constant mean curvature H. Suppose in addition that M has two distinct principal curvatures l and m with multiplicities (n - 1) and 1, respectively. Let c = ±1 be the sign of the difference l - m. Then

with equality only if M is an H(r)-torus^n-1(r) × ¹().

2 PRELIMINARIES

Let M be a compact hypersurface with constant mean curvature H immersed in an (n + 1)-dimensional unit sphere ⁿ⁺¹. Choose a local orthornormal frame field E₁, ...,E_n in a neighborhood U of M and let w₁, ..., w_n be its dual coframe. As is well known, there are smooth 1-forms w_ij on U uniquely determined by the equations

The square of the length of the second fundamental form

is given by . Note that h_ij = h_ji and

The covariant derivative of A is given by

where

It is well known that h_ijk is symmetric in all indices.

If F = A - HI is the traceless second fundamental form, then |F|² = |A|² - nH². Observe that |F|²> 0 and equality holds precisely at the umbilic points of M. For that reason, F is also called the total umbilicity tensor of M. We also have the Simons formula (see for instance (Alencar and do Carmo 1994), taking into account the different choice of sign in their definition of F)

From now we will assume that Mⁿ is a compact hypersurface with constant mean curvature having everywhere two distinct principal curvatures l and m with multiplicities n - k and k, respectively. First, we need a classical result of Otsuki (Otsuki 1970).

PROPOSITION 5. Let M be a hypersurface in ⁿ⁺¹such that the multiplicities of its principal curvatures are constant. Then the distribution D_l of the space of principal vectors corresponding to each principal curvature l is completely integrable. In particular, if the multiplicity of a principal curvature is greater than 1, then this principal curvature is constant on each of the integral leaves of the corresponding distribution of the space of its principal vectors.

A consequence of this result is the following lemma:

LEMMA 6. Let Mⁿ be a compact oriented hypersurface in ⁿ⁺¹with constant mean curvature and two principal curvatures l and m, with multiplicities n - k and k respectively. If 1 < k < n - 1, then M is isometric to ^n-k(r) × ^k().

PROOF. We may choose a local orthonormal basis E₁, ..., E_k, E_k+1, ..., E_n such that for 1 < i < k and k + 1 < j < n,

By Proposition 5 above, we have E_i(l) = E_j(m) = 0 for i < k and k + 1 < j. As

is a constant function, it follows that

Therefore m and l are constant and M is an isoparametric hypersurface. Note that since M is compact, then M is isometric to ^n-k(r) × ^k().

We should mention that if H = (1/n)tr A is the mean curvature of a hypersurface M in ⁿ⁺¹ with second fundamental form A, then l is an eigenvalue of A if and only if = l - H is an eigenvalue of the traceless second fundamental form F = A - HI. Those eigenvalues have the same multiplicities. In the following lemma we are going to evaluate the Laplacian of ln|F|. It turns out that D ln|F| depends on the Alencar-do Carmo polynomial

LEMMA 7. Let M be a compact oriented hypersurface with constant mean curvature H immersed in ⁿ⁺¹having two principal curvatures l and m, with multiplicities n-1 and 1 respectively. Then

where c = ±1 is the sign of the difference l - m.

PROOF. Since M has two distinct principal curvatures at each point, it follows that M has no umbilical points. In particular |F| > 0. Note that

and

which implies that

Using now the Simons formula (7), one gets

That is,

Note that the eigenvalues = l - H and = m - H of F have multiplicities n - 1 and 1, respectively, and are related by = -(n - 1) . Observe that

and

so that

where c = ±1 is the sign of the difference l - m. Moreover,

Therefore, using this into (9), one has

LEMMA 8. Let Mⁿ, n > 3, be a compact oriented hypersurface with constant mean curvature H immersed in ⁿ⁺¹having two principal curvatures l and m with multiplicities n - 1 and 1 respectively, then

PROOF. Let E₁, ..., E_n be a local orthornomal frame such that F(E_n) = E_n and

for 1 < i < n - 1. Since H is a constant function and F = A - HI, then

From

and Proposition 5, it follows that for all 1 < i < n - 1. In particular,

and

By choosing j ¹ k, 1 < j < n - 1 we also have a direct proof that

for all k ¹ n. We also note that

and

Finally,

Therefore, since Ñl = dl(E_n)E_n, one gets

Recall that |F|² = n(n - 1) ² with = l - H. Since H is constant it then follows that

Therefore

3 PROOF OF THE MAIN RESULTS

PROOF OF THEOREM 3: Theorem 3 easily follows from Lemma 7 and Lemma 8. In fact, we get

Integrating now over M we conclude that

This finishes the proof of Theorem 3.

As a consequence of this result, we obtain a new characterization of the H(r)-tori ^n-1(r) × ¹().

PROOF OF COROLLARY 4: By the integral formula in Theorem 3 we have that

In the equality, we have

Then |Ñ ln|F||² = 0 and |F| is a constant function. Therefore

and c|F| is a root of the equation P_H(x) = 0. Even more, by Lemma 8 we also know that ÑF = ÑA = 0, which means that M is isoparametric with two constant distinct principal curvatures. In fact, as in the proof of Lemma 8, consider E₁, ..., E_n a local orthornomal frame such that A(E_n) = mE_n and A(E_i) = lE_i for 1 < i < n - 1. Then

implies that l, and hence m, is constant. Finally, M being compact, it follows that M is isometric to an H(r)-torus ^n-1(r) × ¹().

ACKNOWLEDGMENTS

The second and third authors would like to thank the Departamento de Matematicas at Universidad de Murcia for the hospitality. The third author was partially supported by CAPES BEX 032402/7.

PENG C AND TERNG C. 1983. Minimal hypersurfaces of sphere with constant scalar curvature. Ann Math Stud 103: 177-198.

Manuscript received on February 26, 2004; accepted for publication on May 5, 2004;

presented by JOÃO LUCAS M. BARBOSA

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Publication Dates

• Publication in this collection
  20 Aug 2004
• Date of issue
  Sept 2004

History

• Accepted
  05 May 2004
• Received
  26 Feb 2004