Print version ISSN 0001-3765
An. Acad. Bras. Ciênc. vol.76 no.3 Rio de Janeiro Sept. 2004
Luis J. AlíasI; Sebastião C. de AlmeidaII; 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
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.
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.
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 < BH, where BH is a constant that depends only on H and n, then either |F|2 = 0 or |F|2 = BH. 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 = BH. 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 > CH, where
Then H is constant, |F|2 = CH 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+1 with 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 PH is the Alencar-do Carmo polynomial
where c = ±1 is the sign of the difference l - m.
The polynomial PH 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 BH found by them in that paper is given precisely as the square of the positive root of PH (a constant that depends only on H and n). For that reason we will refer to PH 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+1 with 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().
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, ...,En in a neighborhood U of M and let w1, ..., wn be its dual coframe. As is well known, there are smooth 1-forms wij on U uniquely determined by the equations
The square of the length of the second fundamental form
is given by . Note that hij = hji and
The covariant derivative of A is given by
It is well known that hijk 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+1 such that the multiplicities of its principal curvatures are constant. Then the distribution Dl 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+1 with 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, ..., Ek, Ek+1, ..., En such that for 1 < i < k and k + 1 < j < n,
By Proposition 5 above, we have Ei(l) = Ej(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+1 having 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
which implies that
Using now the Simons formula (7), one gets
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
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+1 having two principal curvatures l and m with multiplicities n - 1 and 1 respectively, then
PROOF. Let E1, ..., En be a local orthornomal frame such that F(En) = En and
for 1 < i < n - 1. Since H is a constant function and F = A - HI, then
and Proposition 5, it follows that for all 1 < i < n - 1. In particular,
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
Therefore, since Ñl = dl(En)En, one gets
Recall that |F|2 = n(n - 1) 2 with = l - H. Since H is constant it then follows that
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 PH(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, ..., En a local orthornomal frame such that A(En) = mEn and A(Ei) = lEi 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().
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.
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Aldir Brasil Jr.
Manuscript received on February 26, 2004; accepted for publication on May 5, 2004;
presented by JOÃO LUCAS M. BARBOSA