Hypersurfaces with Constant Mean Curvature in Spheres

HYPERSURFACES WITH CONSTANT MEAN CURVATURE IN SPHERES

HILARIO´ ALENCAR AND MANFREDO DO CARMO

Abstract. Let M n be a compact hypersurface of a sphere with constant mean curvature H. We introduce a tensor φ, related to H 2 and to the second fundamental form, and show that if |φ| ≤ BH , where BH 6= 0 is a number depending only on H and n, then either 2 2 n 2 |φ| ≡ 0 or |φ| ≡ BH . We also characterize all M with |φ| ≡ BH .

Mathematics Subject Classiﬁcation (1991). Primary 53C42; Secondary 53A10. Key words: Constant mean curvature, spheres, minimal surfaces, totally umbilic, tori.

1. Introduction Let M n be an n-dimensional orientable manifold and let f : M n −→ n+1 n+2 S (1) ⊂ R be an immersion of M into the unit (n + 1)-sphere n+1 n+2 S (1) of the euclidean space R . Choose a unit normal ﬁeld η along f, and denote by A : TpM −→ TpM the linear map of the tangent space TpM, at the point p ∈ M, associated to the second fundamental form of f along η, i.e.,

hAX, Y i = h∇X Y, ηi, where X and Y are tangent vector ﬁelds on M and ∇ is the connection of Sn+1(1). A is a symmetric linear map and can be diagonalized in an orthonormal basis {e1, . . . , en} of TpM, i.e., Aei = kiei, i = 1, . . . , n. 1 P 2 We will denote by H = n i ki the mean curvature of f and by |A| = P 2 i ki . When f is minimal (H ≡ 0) the following gap theorem is well known. Theorem 1.1. Let M n be compact and f : M n −→ Sn+1(1) be a minimal hypersurface. Assume that |A|2 ≤ n, for all p ∈ M. Then: (i) Either |A|2 ≡ 0 (and M n is totally geodesic) or |A|2 ≡ n. 1 2 Hil´arioAlencar and Manfredo do Carmo

(ii) |A|2 ≡ n if and only if M n is a Cliﬀord torus in Sn+1(1), i.e., n n n M is a product of spheres S 1 (r1) × S 2 (r2), n1 + n2 = n, of appropriate radii. Remark 1. The sharp bound (i) is due to Simons [10]. The character- ization given in (ii) was obtained independently by Chern, do Carmo, and Kobayashi [4] and Lawson [5]. The result in (ii) is local. Attemps have been made to extend the above result to hypersurfaces with constant mean curvature H (see, e.g., Okumura [7]), but as far as we know no sharp bound has yet found. The purpose of this paper is to describe such a sharp bound and characterize the hpersurfaces that appear when the bound is reached. For that, it is convenient to deﬁne a linear map φ : TpM −→ TpM by hφX, Y i = HhX,Y i − hAX, Y i. It is easily checked that trace φ = 0 and that 1 X |φ|2 = (k − k )2, i, j =, . . . , n, 2n i j i,j so that |φ|2 = 0 if and only if M is totally umbilic. It turns out that φ is the natural object to use when extending the above theorem to constant mean curvature. In fact, Theorem 1.2 below can be proved. n+1 We need some notation. An H(r)-torus in S (1)√ is obtained by n−1 n 1 2 considering the standard immersions S (r) ⊂ R , S ( 1 − r2) ⊂ R , 0 < r < 1, where the value within the parentheses denotes the ra- dius of the corresponding√ sphere, and taking the product immersion n−1 1 n 2 S (r) x S ( 1 − r2) ,→ R x R . By the choices made, the H(r)- torus turns out to be contained in Sn+1(1) and has principal curvatures given, in some orientation, by √ 1 − r2 r (1.1) k1 = ... = kn−1 = , kn = −√ , r 1 − r2 or the symmetric of these values for the opposite orientation. Let M n be compact and orientable, and let f : M n −→ Sn+1(1) have constant mean curvature H; choose an orientation for M such that H ≥ 0. For each H, set

2 n(n − 2) 2 PH (x) = x + Hx − n(H + 1), pn(n − 1) Hypersurfaces With Constant Mean Curvature in Spheres 3 and let BH be the square of the positive root of PH (x) = 0. Notice that for H = 0, B0 = n. 2 Theorem 1.2. Assume that |φ| ≤ BH for all p ∈ M. Then: 2 2 (i) Either |φ| ≡ 0 (and M is totally umbilic) or |φ| ≡ BH . 2 (ii) |φ| ≡ BH if and only if: (a) H = 0 and M n is locally a Cliﬀord torus. (b) H 6= 0, n ≥ 3, and M n is locally an H(r)-torus with r2 < n−1 n . (c) H 6= 0, n = 2, and M n is locally an H(r)-torus, for any n−1 r 6= n . Remark 2. As it will be seen in the proof, part (ii) of Theorem 1.2 is again a local result. Remark 3. It is an interesting fact that not all H(r)-torus appear in the equality case for n ≥ 3, but only those for which r2 < (n − 1)/n (it can be checked that if we orient those H(r)-torus for which r2 > (n − 1)/n 2 in such a way that H ≥ 0, then |φ| > BH ). This has to do with the fact that the term which contains H in the equation PH (x) = 0 vanishes when n = 2. Thus, if H 6= 0, the equation deﬁning BH is invariant by a change of orientation if and only if n = 2. Remark 4. In the minimal case, Theorem 1.1 can be extended to higher codimensions (see [4]). In her doctoral dissertation of IMPA, Walcy Santos has also been able to extend Theorem 1.2 to higher codimensions (for the precise statement in this case, see [9]).

2. Proof of Theorem 1.2 We ﬁrst compute the Laplacian ∆φ of φ. We ﬁrst observe that given a Riemannian manifold M and a symmetric linear map on the tangent spaces of M that satisfy formally the Codazzi equation, Cheng and Yau [3] have already computed such a Laplacian. This turns out to be the case for φ, and the result of [3] in our context can be described as follows. Let {e1, . . . , en} be an orthonormal frame which diagonalizes φ at each point of M, i.e., φei = µiei, and let ∇ be the inducced connection on M. Then ([3], p.198)

1 1 X (2.1) ∆|φ|2 = |∇φ|2 + µ µ (trφ) + R (µ − µ )2, 2 i i ii 2 ijij i j i,j 4 HilárioAlencar and Manfredo do Carmo where Rijij is the sectional curvature of the plane {ei, ej}. We first compute the last term on the right-hand side of (2.1). By the definition of φ, µi = H − ki and, by Gauss’s formula, 2 Rijij = 1 + kikj = 1 + µiµj − H(µi + µj) + H . We now use a result of Nomizu and Smyth ([6], p.372) which implies, since trφ = 0, that !2 1 X X X (1 + µ µ )(µ − µ )2 = n iµ2 − µ2 . 2 i j i j i i i,j i P 2 2 Therefore, since i,j(µi − µj) = 2n|φ| , we obtain

(2.2) 1 P R (µ − µ )2 = P iµ2 − P µ22 2 i,j ijij i j i i i

H H2 − P (µ + µ )(µ − µ )2 + P (µ − µ )2 2 i,j i j i j 2 i,j i j H = n|φ|2 − |φ|4 + nH2|φ|2 − P (µ + µ )(µ − µ )2. 2 i,j i j i j P On the other hand, since i,j µi = 0, it is easily checked that 1 X X (2.3) R (µ − µ )2 = n µ3. 2 ijij i j i i,j i It follows from (2.2) and (2.3) that (2.1) can be written as 1 X (2.4) ∆|φ|2 = |∇φ|2 − |φ|4 + n|φ|2 + nH2|φ|2 − nH µ3. 2 i i P 3 We want to estimate i µi .. For that, we use the following lemma, the inequality case of which is stated in Okumura [7]. P Lemma 1. Let µi, i = 1, . . . , n be real numbers such that i µi = 0 and P 2 2 i µi = β , where β = const ≥ 0. Then n − 2 3 X 3 n − 2 3 −p β ≤ µi ≤ p β , n(n − 1) i n(n − 1) and equality holds in the right-hand (left-hand) side if and only if (n−1) of the µi’s are nonpositive and equal ((n − 1) of the µi’s are nonnegative and equal). Hypersurfaces With Constant Mean Curvature in Spheres 5

Proof. We can assume that β > 0, and use the method of Lagrange’s P 3 multipliers to ﬁnd the critical points of g = i µi subject to the con- P P 2 2 ditions: i µi = 0, i µi = β . It follows that the critical points are given by the values of µithat satisfy the quadratic equation

µi − λµi − α = 0, i = 1, . . . , n.

Therefore, after reenumeration if necessary, the critical points are given by:

µ1 = µ2 = ... = µp = a > 0, µp+1 = µp+2 = ... = µn = b < 0.

Since, at the critical points,

2 X 2 2 2 β = µi = pa + (n − p)b , i

X 0 = µi = pa − (n − p)b, i

X 3 3 3 g = µi = pa + (n − p)b , i we conclude that n − p p n − p p a2 = β2, b2 = β2, g = a − b β2. pn (n − p)n n n It follows that g decreases when p increases. Hence g reaches a maximum when p = 1, and the maximum of g is given by

a3 − (n − 1)b3 = ((n − 1)b)3 − (n − 1)b3 = (n − 2)n(n − 1)b2b

n − 2 = β3. pn(n − 1)

Since g is symmetric, this proves the lemma. Remark 5. For later use, it is convenient to observe from the proof that the equality holds in the right-hand side if and only if (n − 1) 1/2 µi’s are of the form −b = −(1/n(n − 1)) β and the remaining on is a = ((n − 1)/n)1/2β. 6 Hil´arioAlencar and Manfredo do Carmo

We return to the proof o Theorem 1.2. By using Lemma 1 in (2.4), we obtain n(n − 2) 1 ∆|φ|2 ≥ |∇φ|2 − |φ|4 + n(H2 + 1)|φ|2 − H|φ|3 2 pn(n − 1) ! n(n − 2) = |∇φ|2 + |φ|2 −|φ|2 − H|φ| + n(H2 + 1) . pn(n − 1) Integrating both sides of the above inequality, using Stokes’ theorem and the hypothesis, we conclude that Z Z ! 2 2 2 n(n − 2) 2 0 ≥ |∇φ| + |φ| −|φ| − p H|φ| + n(H + 1) ≥ 0. M M n(n − 1) 2 2 2 Thus |∇φ| ≡ 0 and either |φ| ≡ 0 or |φ| ≡ BH . This proves part (i) of Theorem 1.2. 2 We now consider part (ii). Notice ﬁrst that if |φ| ≡ BH , the right- hand side of inequality 6 vanishes identically irrespective of the com- pactness of M. Since this is all that we will use, the remaining part of the argument is local. If H = 0, the theorem reduces to Theorem 1.1 which gives (ii)(a). If H 6= 0, we conlude that ∇φ = 0 and that equality holds in the right-hand side of Lemma 1. It follows that ki = const and (n − 1) of the ki’s are equal (see, e.g., [4], p.67). After reenumeration if necessary, we can assume that

k1 = k2 = ... = kn−1, k1 6= kn, ki = const. In this situation, if n ≥ 3, a theorem of do Carmo and Dajczer ([2], p. 701) implies that M n is (contained in) a rotation hypersurfaces of Sn+1(1) obtained by rotating a curve of constant curvature. It follows that M n is an H(r)-torus. To identify which H(r)-tori do appear, we ﬁrst observe that the equality case of Lema 1 gives (with the enumeration above): s rn − 1 1 µ = |φ|, µ = µ = ... = µ = − |φ|. n n 1 2 n−1 n(n − 1) Thus ! s ! rn − 1 1 k k = H − |φ| H + |φ| , n 1 n n(n − 1) Hypersurfaces With Constant Mean Curvature in Spheres 7

2 hence, since |φ| ≡ BH ,

2 2 n(n − 2) nknk1 = nH − |φ| − H|φ| = −n, pn(n − 1) that is, nknk1 = 1. On the other hand, from k + (n − 1)k k = H − µ = n 1 − µ , n n n n we conclude that

(n − 1)kn − (n − 1)k1 = −nµn, and, since µn > 0, we obtain that kn < k1. Because knk1 = −1, this implies that kn < 0. It follows that the oriented H(r)-torus selected by the equality case of Lemma 1 is given by (1.1). Since its mean curvature (n − 1) − nr2 H = √ nr 1 − r2 is positive, we must have r2 < (n − 1)/n. This completes the proof of case (b) in (ii). To prove ﬁnally the case (ii)(c), we observe that M 2 ⊂ S3(1) is an isoparametric surface in S3(1) which is known to be either totally umbilic or an H(r)-torus. Since |φ|2 6= 0, M 2 is an H(r)-torus. By the above argument, we see that k2k1 = −1. Now, however, because the equality case of Lemma 1 gives no additional information, we can have both cases: k2 > 0, k1 < 0 and k2 < 0, k1 > 0. Thus, the (positive) mean curvature can be either (n − 1) − nr2 nr2 − (n − 1) H = √ or H = √ , n = 2, nr 1 − r2 nr 1 − r2 2 n−1 and all r 6= n will occur. This concludes the proof of (ii)(c) and of the theorem.

3. Further remarks Theorem 1.2 raises the following question: Consider the set of hypersurfaces of Sn+1(1) with H = const and |φ| = const . Is the set of values of |φ| discrete? For minimal hypersurfaces, this question was raised in [4], and even in this simple case it was shown to be a hard qustion. For n = 3 and H ≡ 0, a signiﬁcant contribution was given by Peng and Terng [8] who showed that if 3 < |φ|2 ≤ 6, |φ| = const, then |φ|2 = 6 and M 3 is a minimal isoparametric hypersurface of S4(1) with three distinct principal curvatures. 8 Hil´arioAlencar and Manfredo do Carmo

The result of Peng and Terng was extended to hypersurfaces of S4(1) with constant mean curvature H by Almeida and Brito [1]. They proved that if |φ|2 = const and |φ|2 ≤ 6 + 6H2, then M 3 is an isoparametric hypersurface of S4(1) with constant mean curvature H; furthermore, if 4+6H2 ≤ |φ|2 ≤ 6+6H2, then |φ|2 = 6+6H2 and M 3 has three distinct principal curvatures. The result of Almeida and Brito solves the above question for n = 3 and |φ|2 ≤ 6 + 6H2 and also throws some light on what happens to 2 2 the H(r)-tori when H 6= 0 and r > 3 : they are all in the interval 2 2 BH < |φ| < 4 + 6H (cf. Remark 3).

Acknowledgments We want to thank Walcy Santos for a critical reading and the referee for valuable suggestions.

References [1] S. de Almeida and F. Brito, Closed 3-dimensional hypersurfaces with constant mean curvature and constant scalar curvature, Duke Math. J. 61 (1990), 195– 206. [2] M. do Carmo and M. Dajczer, Rotation hypersurfaces in spaces of constant curvature, Trans. Amer. Math. Soc. 277 (1983) 685-709. [3] S. Y. Cheng and S. T. Yau, Hypersurfaces with constant scalar curvature, Math. Ann. 225 (1977), 195–204. [4] S. S. Chern, M. do Carmo, and S. Kobayashi, Minimal submanifolds of a sphere with second fundamental form of constant length, Functional Analysis an Related Fields (F. Browder, ed.), Springer-Verlag, Berlin, 1970, pp. 59-75. [5] B. Lawson, Local rigidity theorems for minimal hypersurfaces, Ann. of Math. (1) 89 (1969), 187-197. [6] K. Nomizu and B. Smyth, A formula of Simon’s type and hypersurfaces with constant mean curvature, J. Diﬀerentil Geom. 3 (1969), 367-377. [7] M. Okumura, Hypersurfaces and a pinching problem on the second fundamental tensor, Amer. J. Math. 96 (1974), 207-213. [8] C. K. Peng and C. L. Terng, Minimal hypersurfaces of spheres with constant scalar curvature, Seminar on Minimal Submanifolds (E. Bombieri, ed.), Ann. of Math. Stud., no. 103, Princeton Univ. Press, Princeton, NJ, 1983, pp. 177-198. [9] W. Santos, Submanifolds with parallel mean curvature vector is spheres, An. Acad. Bras. Ciˆenc. 64 (1992), 215-219. [10] J. Simons, Minimal varieties in Riemannian manifolds, Ann. of Math. (2) 88 (1968), 62-105.

HilárioAlencar Universidade Federal de Alagoas, Instituto de Matemática,57072-900, Maceió-AL, Hypersurfaces With Constant Mean Curvature in Spheres 9

Brasil. Email: [email protected]

Manfredo do Carmo IMPA, Estrada D. Castorina, 110 - Jd. Botˆanico, 22460-320, Rio de Janeiro, R.J.- Brasil. Email: [email protected]