Hypersurfaces with constant mean curvature and two principal curvatures in S n + 1

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) and Wang (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.


INTRODUCTION
Let M be a compact minimal hypersurface of the (n + 1)-dimensional unit Euclidean sphere S 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 andLawson (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 S n+1 having distinct principal curvatures of multiplicities k and m = n − k ≥ 2 are locally product of spheres of the type S m (c 1 ) × S n−m (c 2 ), and he constructed examples of compact minimal hypersurfaces in S 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 = A − H I, the so called traceless second fundamental form, Alencar and do Carmo (Alencar and do Carmo 1994) proved that if M n is compact with constant mean curvature H and | | 2 ≤ B H , where B H is a constant that depends only on H and n, then either /n are the only hypersurfaces with constant mean curvature H and | | 2 = B H .These results do not characterize the other tori 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 S n−1 (r) × S 1 ( √ 1 − r 2 ) with r 2 ≥ (n − 1)/n.More precisely: Theorem 1 ( Barbosa et al. 2003).Let M be a compact oriented hypersurface immersed in the sphere S n+1 , with two distinct principal curvatures λ and µ with multiplicities 1 and n − 1, respectively.Suppose in addition that n ≥ 3 and | | 2 ≥ C H , where .
Recently, Perdomo (Perdomo 2004) and Wang (Wang 2003) independently obtained the following integral formula for compact minimal hypersurfaces with two principal curvatures immersed in S n+1 .
Theorem 2 (Perdomo 2004, Wang 2003).Let M n be a compact minimal hypersurface immersed in S n+1 .If M has two principal curvatures everywhere, then if λ 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 S 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 . Explicitly, we have the following result.
Theorem 3. Let M be a compact oriented hypersurface immersed in the sphere S n+1 with constant mean curvature H . Suppose in addition that M has two distinct principal curvatures λ and µ with multiplicities (n − 1) and 1, respectively.If = A − H I is the traceless second fundamental form of M and P H is the Alencar-do Carmo polynomial where c = ±1 is the sign of the difference λ − µ.
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 S n+1 with constant mean curvature H. Suppose in addition that M has two distinct principal curvatures λ and µ with multiplicities (n − 1) and 1, respectively.Let c = ±1 be the sign of the difference λ − µ.

PRELIMINARIES
Let M be a compact hypersurface with constant mean curvature H immersed in an (n + 1)dimensional unit sphere S n+1 .Choose a local orthornormal frame field E 1 , ..., E n in a neighborhood U of M and let ω 1 , ..., ω n be its dual coframe.As is well known, there are smooth 1-forms ω ij on U uniquely determined by the equations The square of the length of the second fundamental form The covariant derivative of A is given by where Observe that | | 2 ≥ 0 and equality holds precisely at the umbilic points of M. For that reason, 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 ) From now we will assume that M n is a compact hypersurface with constant mean curvature having everywhere two distinct principal curvatures λ and µ 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 S n+1 such that the multiplicities of its principal curvatures are constant.Then the distribution D λ of the space of principal vectors corresponding to each principal curvature λ 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 M n be a compact oriented hypersurface in S n+1 with constant mean curvature and two principal curvatures λ and µ, with multiplicities n − k and k respectively.
Proof.We may choose a local orthonormal basis E 1 , . . ., E k , E k+1 , . . ., E n such that for 1 By Proposition 5 above, we have Therefore µ and λ are constant and M is an isoparametric hypersurface.Note that since M is compact, then M is isometric to We should mention that if H = (1/n)trA is the mean curvature of a hypersurface M in S n+1 with second fundamental form A, then λ is an eigenvalue of A if and only if λ = λ − H is an eigenvalue of the traceless second fundamental form = A − H I .Those eigenvalues have the same multiplicities.In the following lemma we are going to evaluate the Laplacian of ln | |.It turns out that ln | | depends on the Alencar-do Carmo polynomial Lemma 7. Let M be a compact oriented hypersurface with constant mean curvature H immersed in S n+1 having two principal curvatures λ and µ, with multiplicities n − 1 and 1 respectively.Then where c = ±1 is the sign of the difference λ − µ.
Proof.Since M has two distinct principal curvatures at each point, it follows that M has no umbilical points.In particular | | > 0. Note that Using now the Simons formula (7), one gets Note that the eigenvalues λ = λ − H and µ = µ − H of have multiplicities n − 1 and 1, respectively, and are related by µ = −(n − 1) λ.Observe that where c = ±1 is the sign of the difference λ − µ.Moreover, Therefore, using this into (9), one has Lemma 8. Let M n , n ≥ 3, be a compact oriented hypersurface with constant mean curvature H immersed in S n+1 having two principal curvatures λ and µ with multiplicities n − 1 and 1 respectively, then and Proposition 5, it follows that for all 1 ≤ i ≤ n − 1.In particular, and By choosing j = k, 1 ≤ j ≤ n − 1 we also have a direct proof that for all k = n.We also note that and Finally, Therefore, since ∇λ = dλ(E n )E n , one gets Recall that | | 2 = n(n − 1) λ 2 with λ = λ − H . Since H is constant it then follows that