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A Hopf Theorem for Ambient Spaces of Dimensions Higher Than Three

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A Hopf Theorem for Ambient Spaces of Dimensions Higher Than Three j. differential geometry 84 (2010) 1-17 A HOPF THEOREM FOR AMBIENT SPACES OF DIMENSIONS HIGHER THAN THREE Hilario´ Alencar, Manfredo do Carmo & Renato Tribuzy Abstract 2 n n We consider surfaces M immersed in Ec R, where Ec is a simply connected n-dimensional complete Riemannian× manifold with constant sectional curvature c = 0, and assume that the mean curvature vector of the immersion is6 parallel in the normal bundle. We consider further a Hopf-type complex quadratic form Q on M 2, where the complex structure of M 2 is compatible with the induced metric. It is not hard to check that Q is holomorphic (see [3], p.289). We will use this fact to give a reasonable description n of immersed surfaces in Ec R that have parallel mean curvature vector. × 1. Introduction A beautiful result on surfaces M 2 immersed in a 3-dimensional eu- clidean space R3 was obtained by H. Hopf in 1951 [6] and states that if M 2 is homeomorphic to a sphere and has constant mean curvature H, then M 2 is totally umbilic, hence isometric to a round sphere. The basic idea of Hopf’s proof is to introduce a complex quadratic form α˜ in M 2 (in the complex structure of M 2 determined by its induced metric) and prove thatα ˜ is holomorphic if H = constant. Hopf’s theo- rem was extended by Chern [4] to surfaces immersed in a 3-dimensional 3 Riemannian manifold Mc (we use superscripts to denote dimensions) with constant sectional curvature c and, recently, for surfaces in simply- connected, homogeneous 3-dimensional Riemannian manifolds with a 4-dimensional group of isometries (Abresch and Rosenberg [1], [2]). It is then natural to look for higher dimensional ambient spaces in which a Hopf-type theorem holds. 2 n R In this paper, we study the case where x: M Ec is a surface immersed in the product Riemannian manifold of→ a simply-con× nected n n-dimensional Riemannian manifold Ec of constant sectional curvature c = 0 with the euclidean line R. We assume that the mean curvature 6 The authors are partially supported by C.N.Pq. Received 05/31/2008. 1 2 H. ALENCAR, M. DO CARMO & R. TRIBUZY vector H is parallel in the normal bundle. By analogy with [1], introduce the real quadratic form in M 2 = M Q(X,Y ) = 2 α(X,Y ),H c X,ξ Y,ξ , h i− h ih i where X, Y are tangent vectors in M, α is the second fundamental form of M taking values in the normal bundle of M, and ξ is the unit tangent vector of R. Let (u, v) be isothermal parameters in M, and z = u + iv the corresponding complex parameter. Set 1 1 dz = (du + idv), dz¯ = (du idv), √2 √2 − and 1 ∂ ∂ 1 ∂ ∂ Z = i , Z = + i √2 ∂u − ∂v √2 ∂u ∂v · In a previous paper [3], we considered this situation and proved that the (2, 0)-part Q(2,0) = ψ dzdz of Q is holomorphic. This means that the complex function Q(2,0)(Z, Z) = 2 α(Z, Z),H c ξ, Z 2 = ψ h i− h i is holomorphic. In the present paper, we want to use the fact that Q(2,0) is holo- morphic to give a reasonable description of which immersed surfaces in n R Ec have parallel mean curvature vector. All manifolds are connected and× oriented. Our first result shows, under no global hypothesis on M, that either n H is an umbilic direction, and our surface is entirely contained in Ec (this reduces our question to a theorem of S.T. Yau; see items (1) and (2) below) or we can reduce the codimension of our immersion to three. n Theorem 1. Let M be a surface and Ec be a Riemannian manifold n R of constant sectional curvature c = 0, and let x: M Ec be an immersion with parallel mean curvature6 vector. Then,→ one× of the following assertions holds: 1) x(M) is a minimal surface in a totally umbilical hypersurface of n Ec . 2) x(M) is a surface with constant mean curvature of a 3-dimensional n totally umbilic or totally geodesic submanifold of Ec . 3) x(M) lies in E4 R. c × To improve the last part of the above result, we add further hypothe- ses on M; this is the subject of the following two theorems. Theorem 2 is where we can be most precise by assuming that M is homeomorphic to a sphere. We will show that either H is an umbilic direction, and we are in the same situation as items (1) and (2) in Theorem 1, or we are in the situation of item (4) below which generalizes the result of Abresch and Rosenberg [1] to immersions of spheres in En R. The proof of (4) is delicate and is the main point of Theorem 2. c × A HOPF THEOREM FOR AMBIENT SPACES 3 Theorem 2. Let M be a compact surface of genus zero and let n R x: M Ec be an immersion of M with parallel mean curvature. Then,→ one of× the following assertions holds: 1) x(M) is a minimal surface of a totally umbilical hypersurface n of Ec . 2) x(M) is a standard sphere of a totally umbilical 3-dimensional n submanifold of Ec . 3 3) x(M) is a standard sphere of Ec . 4 R R6 4) x(M) lies in Ec (possiby with the Lorentz metric), and there exists a plane P such× that⊂ x(M) is invariant for rotations which fix its orthogonal complement. Furthermore, the level curves of the height function p x(p),ξ are circles lying in planes parallel to P . 7→ h i For the next theorem, we assume that M is complete, that its Gauss- ian curvature K 0, and that c< 0. Even for n = 2, Theorem 3 seems to be new. ≥ Theorem 3. Let M be a complete surface with Gaussian curvature n R K 0, and let x: M Ec , c< 0, be an isometric immersion with parallel≥ mean curvature.→ Then× one of the following assertions holds: 1) K 0. 2) x(M≡ ) is a minimal surface of a totally umbilical hypersurface n of Ec . 3) x(M) is a surface with constant mean curvature in a totally um- n bilical 3-dimensional submanifold of Ec . 4 R R6 4) x(M) lies in Ec (with the Lorentz metric), and there exists a plane P such× that⊂ the level lines of the height function p x(p),ξ are curves lying in planes parallel to P . 7→ h i Remark 1. Because the mean curvature vector of our immersion is parallel, the immersion itself is analytic; this means that the functions of 2 n R two real variables that define locally the map x: M Ec are real analytic functions (See, C.B. Morrey Jr. On the analiticity→ × of solutions of analytic nonlinear elliptic systems of partial differential equations, American J. of Math. 80 (1958), 198-237). Such function satisfy a principle of analytic continuation (See J. Dieudonn´e, Foundations of Modern Analysis, Academic Press, 1969, Chapter IX (2.4.2)) which has the following consequence: Let V be an open, connected subset of Rn and let f : V Rk be a real analytic map in V . Let U V be an open subset of Rn.→ If f 0 in U then f 0 in V . Thus an⊂ analytic map cannot vanish in any≡ open set unless≡ it is identically zero. Remark 2. We should mention that holomorphic quadratic forms on surfaces that lie in products of space forms of arbitrary dimensions have been found in the Doctor’s thesis of F.M.A. Vitrio (Fortaleza 2007) and will appear in the joint work of J.H. de Lira and F.M.A. Vitrio: 4 H. ALENCAR, M. DO CARMO & R. TRIBUZY Isometric immersions into Riemannian products, preprint. As far as we know, there is no overlapping with the present paper. Remark 3. An interesting question is to characterize those surfaces for which K 0 in Theorem 3. This seems to be unknown even for the case of a complete≡ M 2 immersed in M 2 R, c = 0. c × 6 Acknowledgement. We want to thank Harold Rosenberg for conver- sations during the preparation of this paper. 2. Proof of Theorem 1 We will start with a few lemmas. n R Lemma 0. Let x: M Ec be an immersion of a surface. Assume that a subbundle L→of the× normal bundle contains the image of the second fundamental form, is parallel in the normal connection and n R that T M L = V is invariant by the curvature tensor R of Ec in the following⊕ sense: Whenever A,B,C V , we have × ∈ e R(A, B)C V. ∈ Then, there exists a totallye geodesic submanifold S En R, with ⊂ c × TpS = V for all p S, so that x(M) S. ∈ ⊂ n R Proof. This follows from ([5], Theorem 2), since Ec , with the Levi- Civitta connection of the product metric, is a reductive× homogeneous space. q.e.d. Remark 4. We recall that for any bundle E B with a connection , we say that a subbundle E′ B, E′ E, is parallel→ if E′ is invariant by∇ the connection . → ⊂ ∇ n R Lemma 1. Let x: M Ec be an immersion of a surface M → × ⊥ with parallel mean curvature. Then for all v T M, AH commutes ∈ with Av ; here Av is the second fundamental form (as a linear map in T M) corresponding to the normal vector v, namely, Av(X),Y = α(X,Y ), v .
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