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Biharmonic Submanifolds in Space Forms

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Biharmonic Submanifolds in Space Forms Symposium Valenciennes 25 Biharmonic submanifolds in space forms ADINA BALMUS¸, STEFANO MONTALDO AND CEZAR ONICIUC ∗ Dipartimento di Matematica, Universita` di Cagliari [email protected], [email protected] Faculty of Mathematics, “Al.I. Cuza” University of Iasi [email protected] Abstract In this paper we outline some recent results concerning the classification of biharmonic submanifolds in a space form. Keywords : harmonic maps, biharmonic maps, minimal submanifolds, biharmonic submanifolds 1 Introduction In [10], even if they took the main interest in harmonic maps, Eells and Sampson also envisaged some generalizations and defined biharmonic maps ϕ : (M,g) → (N,h) between Riemannian manifolds as critical points of the bienergy functional Z 1 2 E2(ϕ) = |τ(ϕ)| vg, 2 M where τ(ϕ) = trace∇dϕ is the tension field of ϕ that vanishes on harmonic maps. The Euler- Lagrange equation corresponding to E2 is given by the vanishing of the bitension field ϕ N τ2(ϕ) = −J (τ(ϕ)) = −∆τ(ϕ) − trace R (dϕ,τ(ϕ))dϕ, where Jϕ is formally the Jacobi operator of ϕ. The operator Jϕ is linear, thus any harmonic map is biharmonic. We call proper biharmonic the non-harmonic biharmonic maps. Although E2 has been on the mathematical scene since the early ’60 (when some of its an- alytical aspects have been discussed) and regularity of its critical points is nowadays a well- developed field, a systematic study of the geometry of biharmonic maps has started only re- cently. In this paper we shall focus our attention on biharmonic submanifolds, i.e. on submanifolds such that the inclusion map is a biharmonic map. The biharmonic submanifolds of a non-positive sectional curvature space that have been con- sidered so far turned out to be all trivial (that is minimal), and the attempts that have been made have led to the following Generalized Chen’s Conjecture: biharmonic submanifolds of a non-positive sectional curva- ture manifold are minimal. On the contrary, the class of proper (non-minimal) biharmonic submanifolds of the sphere is rather rich but a full understanding of their geometry has not yet been achieved. ∗The first author was supported by a INdAM doctoral fellowship, Italy. The third author was supported by the Grant CNCSIS At, 191/2006, Romania. 26 Balmus¸-Montaldo-Oniciuc Our aim is to gather the known results on the classification of biharmonic submanifolds in a space form and to illustrate some recent progress in the theory of biharmonic submanifolds of the sphere. For an up to date bibliography on biharmonic maps we refer the reader to [20]. 2 Biharmonic submanifolds n Let ϕ : M → E (c) be the canonical inclusion of a submanifold M in a constant sectional n curvature c manifold, E (c). The expressions assumed by the tension and bitension fields are, in this case, τ(ϕ) = mH, τ2(ϕ) = −m(∆H − mcH), − n where H, seen as a section of ϕ 1(TE (c)), denotes the mean curvature vector field of M in n − n E (c) and ∆ is the rough Laplacian on ϕ 1(TE (c)). By splitting the bitension field in its normal and tangent components we find that the canon- m n n ical inclusion ϕ : M → E (c) of a submanifold M in an n-dimensional space form E (c) is biharmonic if and only if −∆⊥H − traceB(·,A ·) + mcH = 0, H (1) m 2 2traceA ⊥ (·) + grad(|H| ) = 0, ∇(·)H 2 where A denotes the Weingarten operator, B the second fundamental form, H the mean curva- ture vector field, ∇⊥ and ∆⊥ the connection and the Laplacian in the normal bundle of M in n E (c). n If c ≤ 0, then compact proper biharmonic submanifolds do not exist in E (c). In fact, from a result of Jiang [13], biharmonic maps from a compact manifold to a manifold with non- positive sectional curvature are harmonic. When M is non-compact, then M cannot be proper n biharmonic in E (c), c ≤ 0, provided that the mean curvature is constant [15]. For hypersurfaces, that is n = m + 1, condition (1) takes the simpler form ⊥ 2 ∆ H − (mc − |A| )H = 0, (2) 2Agrad(|H|) + m|H|grad(|H|) = 0. In dimension n = 3, for c ≤ 0, system (2) forces the norm of the mean curvature vector of a 3 surface M2 in E (c) to be constant, which implies the following 3 Theorem 1 ([3, 6]) There exist no proper biharmonic surfaces in E (c), c ≤ 0. For higher dimensional cases it is not known whether there exist proper biharmonic submani- n folds of E (c), n > 3, c ≤ 0, although partial results have been obtained. For instance: n • Every biharmonic curve of R is an open part of a straight line [9]. n • Every biharmonic submanifold of finite type in R is minimal [9]. n • There exist no proper biharmonic hypersurfaces of R with at most two principal curva- tures [9]. Symposium Valenciennes 27 m n • Let M be a pseudo-umbilical submanifold of E (c), c ≤ 0. If m 6= 4, then M is bihar- monic if and only if minimal [3, 9]. 4 • Let M3 be a hypersurface of R . Then M is biharmonic if and only if minimal [14]. n n+1 • A submanifold of S cannot be biharmonic in R [6]. n 2.1 Biharmonic submanifolds of S All the non-existence results described in the previous section do not hold for submanifolds in the sphere. Before we describe some general method to construct biharmonic submanifolds in the sphere lets recall the main examples: p q • ( √1 )× ( √1 ) p+q = n− p 6= q the generalized Clifford torus, S 2 S 2 , 1, , was the first exam- n ple of proper biharmonic submanifold in S [12]; n−1 n • ( √1 ) ⊂ the hypershere S 2 S [2]. For the 3-dimensional unit sphere it is possible to give the full classification of proper bihar- monic submanifolds, as shown by the following Theorem 2 ([2]) 3 a) An arc length parameterized curve γ : I → S is proper biharmonic if and only if it is 1 1 3 either the circle of radius √1 , or a geodesic of the Clifford torus ( √1 ) × ( √1 ) ⊂ 2 S 2 S 2 S with slope different from ±1. 3 2 3 b) A surface M is proper biharmonic in if and only if it is locally a piece of ( √1 ) ⊂ . S S 2 S 2 3 √1 ( √1 ) ⊂ Theorem 2 says that a circle of radius 2 , which is totally geodesic (minimal) in S 2 S , 3 is biharmonic in S . This composition property turned out to be true in any dimension in virtue of the following n−1 n n Proposition 1 ([3]) A minimal submanifold M of S (a) ⊂ S is proper biharmonic in S if and only if a = √1 . 2 This result proved to be quite useful for the construction of proper biharmonic submanifolds in spheres. For instance, using a well known result of Lawson, it implies the existence of closed 4 orientable embedded proper biharmonic surfaces of arbitrary genus in S (see [3]). n A closer look at the biharmonic submanifolds M of S , constructed using Proposition 1, re- 2 veals that they all posses the following features: they are pseudo-umbilical (AH = |H| Id) with parallel mean curvature vector field of norm 1. m1 Nevertheless, it is possible to construct non pseudo-umbilical examples. In fact, if M1 and m n n M 2 m 6= m 1 ( √1 ) 2 ( √1 ) n + n = 2 ( 1 2) are two minimal submanifolds of S 2 and S 2 respectively ( 1 2 n n−1), then M1 ×M2 is a proper biharmonic submanifold in S , which is not pseudo-umbilical, with parallel mean curvature vector field and |H| ∈ (0,1) (see [3]). The value of |H| between 0 and 1 in the previous examples is not a coincidence, in fact it n was proved in [16] that any proper biharmonic constant mean curvature submanifold M in S satisfies |H| ∈ (0,1]. Moreover, if |H| = 1, then M is a minimal submanifold of the hypersphere n−1 n ( √1 ) ⊂ S 2 S . 28 Balmus¸-Montaldo-Oniciuc The forementioned result is related with submanifold of finite type. Lets first recall that an n isometric immersion ϕ : M → R is called of finite type if ϕ can be expressed as a finite sum of n R −valued eigenfunctions of the Beltrami-Laplace operator ∆ of M. When M is compact it is called of k-type if the spectral decomposition of ϕ contains exactly k non-zero terms, excepting the center of mass (see [7]). n n+1 If M is a submanifold of S then M can be seen as a submanifold of R . We say that M is n n+1 n of finite type in S if it is of finite type as a submanifold of R . Denote by ϕ : M → S the n n n+1 n+1 inclusion of M in S and by i : S → R the canonical inclusion. Let φ : M → R , φ = i◦ϕ, n+1 n be the inclusion of M in R . Denoting by H the mean curvature vector field of M in S and n+1 by H0 the mean curvature vector field of M in R we have immediately H0 = H − φ. From [3, Proposition 4.1], we get that τ2(ϕ) = 0 if and only if ∆H0 − 2mH0 + m(|H|2 − 1)φ = 0. (3) From (3), using the Minimal Polynomial Criterion for submanifolds of finite type (see, for example, [6]), we can prove the following m n Theorem 3 ([1]) Let M be a compact constant mean curvature submanifold in S , |H|2 = k.
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