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 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 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 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 , 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. Then M is proper biharmonic if and only if either |H|2 = 1 and M is a 1-type submanifold 2 with eigenvalue√ λ = 2m, or |H| = k ∈ (0,1) and M is a 2-type submanifold with eigenvalues λ1,2 = m(1 ± k).

n Note that all proper biharmonic submanifolds of S with |H| = 1 are 1-type submanifolds in n+1 R , independently on whether they are compact or not.

3 Biharmonic hypersurfaces

The full classification of biharmonic hypersurfaces in a space form is not known and so far only few cases have been studied. The simplest assumption that M is an umbilical hypersurface, i.e. all principal curvatures are equal, does not produce new examples. In fact, if M is a proper m+1 m ( √1 ) biharmonic umbilical hypersurface in S , then it is an open part of S 2 . Moreover, there m+1 m+1 exist no proper biharmonic umiblical hypersurfaces in R or in the hyperbolic space H . Similarly to the case of the Euclidean space (see [9]), the study of proper biharmonic hyper- surfaces with at most two distinct principal curvatures constitutes the next natural step for the classification of proper biharmonic hypersurfaces in space forms. We underline the fact that there exist examples of hypersurfaces with at most two distinct principal curvatures and non-constant mean curvature in any space form. The classification of biharmonic hypersurfaces with at most two distinct principal curvatures relies on the proof that they have constant mean curvature. This property was proved in [9] for the Euclidean case and in [1] for any c.

Theorem 4 ([1]) A biharmonic hypersurface with at most two distinct principal curvatures in m+1 E (c) has constant mean curvature.

As an immediate consequence of Theorem 4 and system (2) we have the following non- existence result

Theorem 5 There exist no proper biharmonic hypersurfaces with at most two distinct principal m+1 m+1 curvatures in R and in H . Symposium Valenciennes 29

The case of the sphere is essentially different. Theorem 4 proves to be the main ingredient for the following complete classification of proper biharmonic hypersurfaces with at most two distinct principal curvatures.

Theorem 6 Let Mm be a proper biharmonic hypersurface with at most two distinct principal m+1 m m m curvatures in . Then M is an open part of ( √1 ) or of 1 ( √1 ) × 2 ( √1 ), m + m = m, S S 2 S 2 S 2 1 2 m1 6= m2.

m+1 Proof. By Theorem 4, the mean curvature of M in S is constant and, from (2), we obtain |A|2 = m. These imply that M has constant principal curvatures. For |H|2 = 1 we conclude m 2 M ( √1 ) |H| ∈ ( , ) M that is an open part of S 2 . For 0 1 we deduce that has two distinct constant principal curvatures. Proposition 2.5 in [17] implies that M is an open part of the product of m1 m2 2 2 two spheres S (a) × S (b), such that a + b = 1, m1 + m2 = m. Since M is biharmonic in n a = b = √1 m 6= m S , from Proposition 1, it follows that 2 and 1 2. We recall that a is called conformally flat if for every point it admits an open neighborhood conformally diffeomorphic to an open set of an Euclidean space. Also, a hypersurface Mm ⊂ Nm+1 which admits a principal curvature of multiplicity at least m − 1 is called quasi-umbilical. For biharmonic hypersurfaces we have the following classification

m m+1 Theorem 7 ([1]) Let M , m ≥ 3, be a proper biharmonic hypersurface in S . The following statements are equivalent

a) M is quasi-umbilical,

b) M is conformally flat,

m 1 m−1 c) M is an open part of ( √1 ) or of ( √1 ) × ( √1 ). S 2 S 2 S 2

3.1 Isoparametric hypersurfaces

m m+1 We recall that a hypersurface M in S is said to be isoparametric of type ` if it has constant principal curvatures k1 > ... > k` with respective constant multiplicities m1,...,m`, m = m1 + m2 + ... + m`. It is known that the number ` is either 1,2,3,4 or 6. For ` ≤ 3 we have the following classification of compact isoparametric hypersurfaces. If ` = 1, then M is totally umbilical.

m1 m2 2 2 If ` = 2, then M = S (r1) × S (r2), r1 + r2 = 1 (see [17]). q If ` = 3, then m1 = m2 = m3 = 2 , q = 0,1,2,3 (see [4]). π Moreover, there exists an angle θ, 0 < θ < ` , such that

(α − 1)π k = cotθ + , α = 1,...,`. (4) α `

Using this classification we can prove the following non-existence result for biharmonic hyper- surfaces with 3 distinct principal curvatures.

Theorem 8 There exist no compact proper biharmonic hypersurfaces of constant mean curva- ture with 3 distinct principal curvatures in the unit Euclidean sphere. 30 Balmus¸-Montaldo-Oniciuc

Proof. First note that a proper biharmonic hypersurface M with constant mean curvature |H|2 = m+1 k in S has constant scalar curvature s = m2(1 + k) − 2m (see [1]). Since M is compact with 3 distinct principal curvatures and constant scalar curvature, from a result of Chang [5], M is m+1 isoparametric with ` = 3 in S . Now, taking into account (4), there exists θ ∈ (0,π/3) such that √ √ π k1 − 3 2π k1 + 3 k1 = cotθ, k2 = cot θ + = √ , k3 = cot θ + = √ . 3 1 + 3k1 3 1 − 3k1 Thus, from Cartan’s classification, the square of the norm of the shape operator is

9k6 + 45k2 + 6 |A|2 = 2q(k2 + k2 + k2) = 2q 1 1 (5) 1 2 3 2 2 (1 − 3k1) and m = 3 · 2q, q = 0,1,2,3. On the other hand, since M is biharmonic of constant mean curvature, from (2), |A|2 = m = 3 · 2q.

6 4 2 The last equation, together with (5), implies that k1 is a solution of 3k1 − 9k1 + 21k1 + 1 = 0 which is an equation with no real roots.

Corollary 1 The only proper biharmonic compact hypersurfaces of constant mean curvature 4 3 1 2 in are the hypersphere ( √1 ) and the torus ( √1 ) × ( √1 ). S S 2 S 2 S 2

m+1 The full classification of proper biharmonic isoparametric hypersurfaces of S is due to Ichiyama, Inoguchi and Urakawa

m+1 Theorem 9 ([11]) A compact isoparametric hypersurface M of S is proper biharmonic if and only if it is one of the following: the hypersphere m( √1 ) or the Clifford torus m1 ( √1 ) × S 2 S 2 m2 ( √1 ), m + m = m, m 6= m . S 2 1 2 1 2

4 Biharmonic submanifolds of codimension greater than one

In this section we shall start a program to classify biharmonic submanifolds of the sphere with n higher codimension. If we assume that M is a pseudo-umbilical submanifold of S , then we soon find strong conditions. The first is

n Theorem 10 ([1]) A biharmonic pseudo-umbilical submanifold of S , m 6= 4, has constant mean curvature.

m m+2 Now if M is a pseudo-umbilical submanifold in S with constant mean curvature, then a m+2 result of Chen [8, p.180] ensures that M is either a minimal submanifold of S or a minimal m+2 hypersurface of a hypersphere of S . Thus, from Proposition 1, we get the following rigidity result

m m+2 Theorem 11 ([1]) A pseudo-umbilical submanifold M of S , m 6= 4, is proper biharmonic m+1 if and only if it is minimal in ( √1 ). S 2

If we replace the condition that M is be pseudo-umbilical with that of being a hypersurface of m+2 a hypersphere in S we have Symposium Valenciennes 31

m m+1 m+2 Theorem 12 ([1]) Let M be a hypersurface of S (a) ⊂ S , a ∈ (0,1). Assume that M is m+1 m+2 not minimal in (a). Then it is biharmonic in if and only if a > √1 and M is open in S S 2 m m+1 ( √1 ) ⊂ (a). S 2 S

Using Theorem 12 we can prove

Theorem 13 A proper biharmonic surface M2 in Sn with parallel mean curvature vector field n−1 is minimal in S ( √1 ). 2

Proof. Chen and Yau proved (see [8, p.106]) that the only non-minimal surfaces with parallel n n−1 mean curvature vector field in S are either minimal surfaces of small hyperspheres S (a) of n n S or surfaces with constant mean curvature in 3-spheres of S . If M is a of a n−1 n (a) a = √1 M small hypersphere S , then it is biharmonic in S if and only if 2 . If is a surface 3 n in a 3-sphere S (a), a ∈ (0,1], of S then we can consider the composition

3 4 n M −→ S (a) −→ S −→ S .

n 4 Note that M is biharmonic in S if and only if it is biharmonic in S . From Theorem 12, for 3 a ∈ ( , ) a = √1 M ( √1 ) a > √1 M 0 1 , we conclude that either 2 and is minimal in S 2 , or 2 and is an 2 2 ( √1 ) a = M ( √1 ) open part of S 2 . For 1, from Theorem 2, also follows that is an open part of S 2 . n−1 M ( √1 ) In all cases is minimal in S 2 . We point out that there exist examples of proper biharmonic constant mean curvature surfaces n n−1 S ( √1 ) in S that are not minimal in 2 . For example, Sasahara, in [18], constructed a proper 5 biharmonic immersion with constant mean curvature ϕ : M2 → S whose position vector field 6 x0 = x0(u,v) in R is given by: √ √ x (u,v) = √1 eiu,ie−iu sin( 2v),ie−iu cos( 2v). 0 2

An easy computation shows that ϕ does not have parallel mean curvature vector field. New examples of proper biharmonic submanifolds of parallel mean curvature vector field were obtained by Zhang in [19]. In particular, he determined all proper biharmonic tori T n+1 = 1 1 2n+1 S (a1) × ··· × S (an+1) in S . We end this paper proposing two conjectures. m+1 Conjecture. The only proper biharmonic hypersurfaces in S are the open parts of hyper- m m m spheres ( √1 ) or of generalized Clifford tori 1 ( √1 ) × 2 ( √1 ), m + m = m, m 6= m . S 2 S 2 S 2 1 2 1 2 n Conjecture. Any biharmonic submanifold in S has constant mean curvature.

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