aa r X i v : . [ m a t h . DG ] S e p BIHARMONIC SUBMANIFOLDS INTO ELLIPSOIDS
S. MONTALDO AND A. RATTO
Abstract.
In this paper we construct proper biharmonic submanifolds into various typesof ellipsoids. We also prove, in this context, some useful composition properties which canbe used to produce large families of new proper biharmonic immersions. Introduction
Harmonic maps are critical points of the energy functional(1.1) E ( ϕ ) = 12 Z M | dϕ | dv g , where ϕ : ( M, g ) → ( N, h ) is a smooth map between two Riemannian manifolds M and N .In analytical terms, the condition of harmonicity is equivalent to the fact that the map ϕ isa solution of the Euler-Lagrange equation associated to the energy functional (1.1), i.e.(1.2) trace ∇ dϕ = 0 . The left member of (1.2) is a vector field along the map ϕ , or, equivalently, a section of thepull-back bundle ϕ − ( T N ): it is called tension field and denoted τ ( ϕ ).A related topic of growing interest deals with the study of the so-called biharmonic maps :these maps, which provide a natural generalisation of harmonic maps, are the critical pointsof the bienergy functional (as suggested by Eells–Lemaire [10]) E ( ϕ ) = 12 Z M | τ ( ϕ ) | dv g . In [11] G. Jiang derived the first variation and the second variation formulas for the bienergy.In particular, he showed that the Euler-Lagrange equation associated to E ( ϕ ) is(1.3) τ ( ϕ ) = − J ( τ ( ϕ )) = −△ τ ( ϕ ) − trace R N ( dϕ, τ ( ϕ )) dϕ = 0 , where J denotes (formally) the Jacobi operator of ϕ , △ is the rough Laplacian on sectionsof ϕ − ( T N ) that, for a local orthonormal frame { e i } mi =1 on M , is defined by(1.4) ∆ = − m X i =1 {∇ ϕe i ∇ ϕe i − ∇ ϕ ∇ Mei e i } , and R N ( X, Y ) = ∇ X ∇ Y − ∇ Y ∇ X − ∇ [ X,Y ] Mathematics Subject Classification.
Key words and phrases.
Biharmonic maps, biharmonic submanifols, ellipsoids, composition properties.Work supported by P.R.I.N. 2010/11 – Variet`a reali e complesse: geometria, topologia e analisi armonica– Italy. s the curvature operator on ( N, h ). We point out that (1.3) is a fourth order semi-linearelliptic system of differential equations. We also note that any harmonic map is an absoluteminimum of the bienergy, and so it is trivially biharmonic. Therefore, a general workingplan is to study the existence of biharmonic maps which are not harmonic: these shall bereferred to as proper biharmonic maps . We refer to [12] for existence results and generalproperties of biharmonic maps.An immersed submanifold into a Riemannian manifold (
N, h ) is called a biharmonic sub-manifold if the immersion is a biharmonic map. In a purely geometric context, B.-Y. Chen[8] defined biharmonic submanifolds M ⊂ R n of the Euclidean space as those with harmonicmean curvature vector field, that is ∆ H = (∆ H , . . . , ∆ H n ) = 0, where H = ( H , . . . , H n ) isthe mean curvature vector as seen in R n and ∆ is the Beltrami-Laplace operator on M . It isimportant to point out that, if we apply the definition of biharmonic maps to immersions intothe Euclidean space, we recover Chen’s notion of biharmonic submanifolds. In this sense,our work can be regarded in the spirit of a generalization of Chen’s biharmonic submanifolds.A general result of Jiang [11] tells us that a compact, orientable, biharmonic submanifold M into a manifold N such that Riem N ≤ N withRiem N ≤ ≥ Biharmonic submanifolds into ellipsoids
We begin with the study of biharmonic submanifolds into Euclidean ellipsoids Q p + q +1 ( c, d )defined as follows: Q p + q +1 ( c, d ) = (cid:26) ( x, y ) ∈ R p +1 × R q +1 = R n : | x | c + | y | d = 1 (cid:27) , where c, d are fixed positive constants. The symmetry of Q p + q +1 ( c, d ) makes it naturalto look for biharmonic generalized Clifford’s tori. More precisely, we shall study isometricimmersions of the following type:(2.1) i : S p ( a ) × S q ( b ) −→ Q p + q +1 ( c, d )( x , . . . , x p +1 , y , . . . , y q +1 ) ( x , . . . , x p +1 , y , . . . , y q +1 ) , here i denotes the inclusion and the radii a, b must satisfy the following condition:(2.2) a c + b d = 1 . In this context, we shall prove the following result:
Theorem 2.1.
Let i : S p ( a ) × S q ( b ) → Q p + q +1 ( c, d ) be an isometric immersion as in (2.1) .If (2.3) a = c pp + q ; b = d qp + q then the immersion is minimal. If (2.3) does not hold and (2.4) a = c cc + d ; b = d dc + d , then the immersion is proper biharmonic. Remark 2.2.
We observe that, interestingly, if c = p and d = q , then we have generalizedminimal Clifford’s tori, but we do not have proper biharmonic submanifolds of the type (2.1).We also point out that, according to Theorem 2.1, the ellipsoid Q ( c, d ) ( p = q = 1 , c = d )admits a proper biharmonic torus, while in S there exists no genus 1 proper biharmonicsubmanifold (see [7]). Proof.
We shall work essentially by using coordinates in R n , suitably restricted to the ellip-soid or to the torus, according to necessity. In particular, the splitting( x, y ) = ( x , . . . , x p +1 , y , . . . , y q +1 )will be used in an obvious way, without further comments. The symbol h , i will denote theEuclidean scalar product (whether in R p +1 , R q +1 or R n will be clear from the context). Weshall use a superscript Q for objects concerning the ellipsoid, while the letter T will appearfor reference to the torus T = S p ( a ) × S q ( b ).We shall need to know the algebraic conditions which ensure that a given vector field istangent either to the torus or to the ellipsoid. More specifically, a vector field W = ( X, Y ) , where X = p +1 X i =1 X i ∂∂x i and Y = q +1 X j =1 Y j ∂∂y j , is tangent to Q p + q +1 ( c, d ) if and only if p +1 X i =1 c x i X i + q +1 X j =1 d y j Y j = 0 . In the same order of ideas, W is tangent to the torus T if and only if(2.5) p +1 X i =1 x i X i = 0 = q +1 X j =1 y j Y j . o end preliminaries, we observe that the vector field(2.6) η Q = η Q | η Q | , where η Q = (cid:18) c x , . . . , c x p +1 , d y , . . . , d y q +1 (cid:19) , represents a unit normal vector field on the ellipsoid Q p + q +1 ( c, d ). Note, for future use, thatthe equality(2.7) | η Q | = a c + b d holds on T . Similarly, the vector(2.8) η T = η T | η T | , where(2.9) η T = (cid:18) c a x , . . . , c a x p +1 , − d b y , . . . , − d b y q +1 (cid:19) , represents a unit normal vector on the torus T viewed as a submanifold of the ellipsoid Q p + q +1 ( c, d ). We also note that(2.10) | η T | = c a + d b on T . In order to compute the tension and the bitension fields, it is convenient to makeexplicit the formulas which will enable us to calculate the relevant covariant derivatives.More precisely, following, for instance, [9], we know that(2.11) ∇ QW W = ∇ R n W W − B Q ( W , W ) , where B Q ( W , W ) denotes the second fundamental form of the ellipsoid Q p + q +1 ( c, d ) into R n . We now need to do some work to make (2.11) more explicit: B Q ( W , W ) = − h ∇ R n W η Q , W i η Q = − h W (cid:18) | η Q | (cid:19) η Q + 1 | η Q | ∇ R n W η Q , W i η Q = − h | η Q | ∇ R n W η Q , W i η Q . (2.12)Next, we compute(2.13) ∇ R n W η Q = 1 c p +1 X i =1 X i ∂∂x i + 1 d q +1 X j =1 Y j ∂∂y j . Finally, using (2.13) into (2.12), we obtain:(2.14) B Q ( W , W ) = − | η Q | (cid:20) c h X , X i + 1 d h Y , Y i (cid:21) η Q , hich in (2.11) yields: ∇ QW W = ∇ R n W W + 1 | η Q | (cid:20) c h X , X i + 1 d h Y , Y i (cid:21) η Q . Now, we are in the right position to proceed to the computation of the tension field τ of ourimmersion (2.1). Indeed, by definition,(2.15) τ = trace B T ( · , · ) , where B T ( W , W ) = − h ∇ QW η T , W i η T = − | η T | h ∇ QW η T , W i η T = − | η T | h ∇ R n W η T , W i η T = − | η T | (cid:20) c a h X , X i − d b h Y , Y i (cid:21) η T . (2.16)Let now X i , i = 1 , . . . , p and Y j , j = 1 , . . . , q , be local orthonormal bases of S p ( a ) and S q ( b ) respectively. By using (2.16) in (2.15) we find: τ = p X i =1 B T (( X i , , ( X i , q X i =1 B T ((0 , Y j ) , (0 , Y j ))(2.17) = − | η T | (cid:20) p c a − q d b (cid:21) η T = λ η T , where, taking into account (2.10), we have set λ = − (cid:20) c a + d b (cid:21) − (cid:20) p c a − q d b (cid:21) . In particular, using (2.2), it is now immediate to conclude that (2.3) is equivalent to theminimality of the immersion.Next, we proceed to the computation of the bitension field τ . To this purpose, we mustapply (1.3) in the case that ϕ = i . We begin with the computation of ∆ τ . It is convenient tochoose a geodesic local orthonormal frame obtained from geodesic local orthonormal frameson each factor of the torus. Under this assumption the terms ∇ Te i e i in the formula (1.4)vanish (note that this simplification is acceptable because we shall not need to computecovariant derivatives of higher order). So the expression for the rough Laplacian (1.4) in ourcontext reduces to:(2.18) ∆ τ = − " p X i =1 ∇ QX i (cid:16) ∇ QX i τ (cid:17) + q X j =1 ∇ QY j (cid:16) ∇ QY j τ (cid:17) , here, to simplify notation, we have written X i for ( X i ,
0) and Y j for (0 , Y j ) . Using (2.11),(2.9) and (2.17) we find: ∇ QX i τ = λ ∇ QX i η T = λ ∇ R n X i η T + λ | η Q | (cid:20) c h X i , η T i + 1 d h , η T i (cid:21) η Q = λ c a X i + λ | η Q | [ 0 ] η Q = λ c a X i . (2.19)Next, using first (2.19), ∇ QX i (cid:16) ∇ QX i τ (cid:17) = λ c a ∇ QX i X i = λ c a (cid:2) ∇ TX i X i + B T ( X i , X i ) (cid:3) = − λ c a | η T | (cid:20) c a h X i , X i i (cid:21) η T , (2.20)where, in order to obtain the last equality, we have used the fact that our orthonormal frameis geodesic and also (2.16). Now, a very similar computation leads us to(2.21) ∇ QY j (cid:16) ∇ QY j τ (cid:17) = − λ d b | η T | (cid:20) d b h Y j , Y j i (cid:21) η T . Putting together (2.18), (2.20) and (2.21) we obtain(2.22) ∆ τ = µ η T , where, taking into account (2.10), we have defined the constant µ as follows:(2.23) µ = λ | η T | (cid:20) p c a + q d b (cid:21) (note that, if (2.3) does not hold, then λ = 0, so that µ = 0 and the immersion is notminimal).By way of summary, the previous computations have led us to the following conclusion:(2.24) τ = − (cid:2) µ η T + trace R Q ( d i, τ ) d i (cid:3) . We have to investigate for which values (if any) of a, b the bitension τ vanishes. In order todeal in an efficient way with the curvature tensor, we shall study the vanishing of normal andtangential components separately. In particular, we shall prove that the normal componentof τ is identically zero if and only if (2.4) holds. The proof of the theorem will then becompleted by the verification that the tangential part of τ vanishes for all values of a and b . So, let us first study whether, for suitable values of a and b , we can have(2.25) h τ , η T i = 0 . rom (2.24) and (2.23) we have: − h τ , η T i = λ " (cid:18) p c a + q d b (cid:19) + p X i =1 h R Q ( X i , τ ) X i , η T i + q X i =1 h R Q ( Y j , τ ) Y j , η T i . Next, we observe that(2.26) h R Q ( X i , τ ) X i , η T i = − h R Q ( X i , τ ) η T , X i i| η T | | η T | = K Q ( X i , η T ) | η T | , where K Q ( X i , η T ) denotes sectional curvature, which (see [9]) can be expressed by means of:(2.27) K Q ( X i , η T ) = h B Q ( X i , X i ) , B Q ( η T , η T ) i − h B Q ( X i , η T ) , B Q ( X i , η T ) i . By using (2.26) in (2.27) and performing a computation which, according to (2.14), uses B Q ( X i , X i ) = − | η Q | c h X i , X i i η Q ,B Q ( η T , η T ) i = − | η Q | (cid:18) c a + d b (cid:19) η Q , (2.28) B Q ( X i , η T ) = 0 , we find:(2.29) h R Q ( X i , η T ) X i , η T i = − | η Q | c h X i , X i i (cid:18) c a + d b (cid:19) . In a very similar fashion we also compute:(2.30) h R Q ( Y j , η T ) Y j , η T i = − | η Q | d h Y j , Y j i (cid:18) c a + d b (cid:19) . Putting together (2.29), (2.30) and (2.24) it is easy to obtain the following conclusion:(2.31) h τ , η T i = − λ (cid:26)(cid:20) p c a + q d b (cid:21) − | η Q | (cid:20) c a + d b (cid:21) h pc + qd i (cid:27) . Now, using (2.7) and (2.2) in (2.31), it is not difficult to check that (2.25) holds if and only if(2.4) is satisfied. At this stage, we can say that the proof of the theorem will be completedif we show that(2.32) h τ , W i = 0for any vector field W which is tangent to the torus. Taking into account (2.24), we see that(2.32) is equivalent to: h trace R Q ( d i, τ ) d i , W i = 0 . Because of (2.17), it is enough to show that h R Q ( X, η T ) X, W i = 0holds if X, W are arbitrary vectors tangent to T . But, by the Gauss equation (see [9]), wededuce:(2.33) h R Q ( X, η T ) X, W i = h B Q ( X, W ) , B Q ( η T , X ) i − h , B Q ( η T , W ) , B Q ( X, X ) i = 0 , where, for the last equality, we have used (2.28). (cid:3) ext, we study biharmonic submanifolds into Euclidean ellipsoids of revolution Q p +1 ( c, d )defined as follows: Q p +1 ( c, d ) = (cid:26) ( x, y ) ∈ R p +1 × R = R n : | x | c + y d = 1 (cid:27) , where c, d are fixed positive constants. In this case, the symmetry of Q p +1 ( c, d ) makesit natural to look for biharmonic hyperspheres. More precisely, we shall study isometricimmersions of the following type:(2.34) i : S p ( a ) × { b } −→ Q p +1 ( c, d )( x , . . . , x p +1 , b ) ( x , . . . , x p +1 , b ) , where i denotes the inclusion and the constants a, b must again satisfy the condition(2.35) a c + b d = 1(note that a is a radius, so it is positive, while the only request on b is: | b | < d ).In this context, we shall prove the following result: Theorem 2.3.
Let i : S p ( a ) × { b } → Q p +1 ( c, d ) be an isometric immersion as in (2.34) . If (2.36) a = c ; b = 0 then the immersion is minimal (this is the case of the equator hypersphere). If (2.37) a = c r cc + d ; b = ± d r dc + d , then the immersion is proper biharmonic.Proof. Again, we shall use a superscript Q for objects concerning the ellipsoid, while theletter S will appear for reference to the hypersphere S = S p ( a ) × { b } . Essentially, the prooffollows the arguments of Theorem 2.1 and most of the calculations can be performed bysetting q = 0 in the formulas above: for this reason, we limit ourselves to point out therelevant differences only. First, let us assume that b = 0 : normal vectors η Q , η Q , η S and η S can be introduced precisely as in (2.6)–(2.10). We also note that, since b = 0, (2.5) impliesthat a tangent vector to S must be of the form(2.38) W = ( X, . Taking into account (2.38) we easily obtain τ = λ η S , where λ = − (cid:20) c a + d b (cid:21) − (cid:20) p c a (cid:21) (note that λ = 0, so that in this case the hypersphere is not minimal).In the computation of ∆ τ only the terms ∇ QX i (cid:16) ∇ QX i τ (cid:17) in (2.18) are relevant: this fact leadsus to the expression(2.39) ∆ τ = µ η S , here now µ = λ | η S | (cid:20) p c a (cid:21) . Also the calculation involving the curvature terms follows the lines above and leads us to(2.40) h τ , η S i = − λ (cid:26)(cid:20) p c a (cid:21) − | η Q | (cid:20) c a + d b (cid:21) h pc i (cid:27) . Now, inspection of (2.40) shows that (2.37) is equivalent to the vanishing of the normal com-ponent of the bitension. Finally, an argument as above shows that the tangential componentof the bitension always vanishes, so ending the case b = 0.In the case that b = 0 we observe that(2.41) η S = (0 , . . . , . Using (2.41) in (2.16) it is easy to conclude that, in this case, the second fundamental formof S vanishes identically, so that the equator hypersphere is totally geodesic and so minimal,a fact which ends the theorem. (cid:3) Composition properties
Our first result is:
Theorem 3.1.
Let i : S p ( a ) → Q p +1 ( c, d ) be a proper biharmonic immersion as in Theorem2.3, and let ϕ : M m → S p ( a ) be a minimal immersion. Then i ◦ ϕ : M m → Q p +1 ( c, d ) is aproper biharmonic immersion.Proof. Let W i , i = 1 , . . . , m , be a local orthonormal frame on M m . To simplify notation, fora tangent vector W to M m , we write W for both dϕ ( W ) and di ( dϕ ( W )) . The compositionlaw for the tension field (see [10]), together with the minimality of ϕ , gives: τ ( i ◦ ϕ ) = m X i =1 ∇ d i ( W i , W i ) + τ ( ϕ )= m X i =1 ∇ d i ( W i , W i )= m X i =1 B S ( W i , W i ) . (3.1)Now, adapting the calculation of (2.16), we have:(3.2) B S ( W i , W i ) = − c a | η S | h W i , W i i η S . Next, using (3.2) in (3.1), we obtain(3.3) τ ( i ◦ ϕ ) = − m c a | η S | η S . n particular, we deduce from (3.3) that i ◦ ϕ is not minimal and we proceed to the compu-tation of the bitension. For convenience, we set ν = m c a | η S | . Using (3.3) we have: τ ( i ◦ ϕ ) = − ∆ M τ ( i ◦ ϕ ) − m X i =1 R Q ( W i , τ ( i ◦ ϕ )) W i = ν " ∆ M η S + m X i =1 R Q ( W i , η S )) W i . (3.4)Next, we study separately the two terms in the right-hand side of (3.4). First, computingas in (2.39) (with p replaced by m ), we find(3.5) ∆ M η S = (cid:20) m | η S | c a (cid:21) η S . Second, using the Gauss equation as in (2.33), we obtain:(3.6) h m X i =1 R Q ( W i , η S )) W i , η S i = − m | η S | | η Q | c (cid:18) c a + d b (cid:19) , and(3.7) h m X i =1 R Q ( W i , η S )) W i , W i = 0for all vector W which is tangent to S . Putting together (3.4)–(3.7) we conclude that τ ( i ◦ ϕ ) is parallel to η S and vanishes if and only if(3.8) (cid:26)(cid:20) c a (cid:21) − | η Q | (cid:20) c a + d b (cid:21) (cid:20) c (cid:21) (cid:27) = 0 . But (3.8) is equivalent to the two conditions (2.37) and (2.35), so that the proof is completed. (cid:3)
In a spirit similar to the previous theorem, we also obtain the following result:
Theorem 3.2.
Let i : S p ( a ) × S q ( b ) → Q p + q +1 ( c, d ) be a proper biharmonic immersion asin Theorem 2.1, and let ϕ : M m → S p ( a ) , ϕ : M m → S q ( b ) be two minimal immersions.Then i ◦ ( ϕ × ϕ ) : M m × M m → Q p + q +1 ( c, d ) is a proper biharmonic immersion.Proof. The proof of this result is a straightforward variant of the arguments of Theorem 3.1and so the details are omitted. (cid:3)
Remark 3.3.
When c = d = 1 the composition properties described in Theorem 3.1 andTheorem 3.2 reduce to those first proved in [6]. It is important to note that all biharmonicsubmanifolds into the ellipsoids constructed using the composition properties have parallelmean curvature vector field. eferences [1] A. Balmu¸s, S. Montaldo, C. Oniciuc. Biharmonic PNMC submanifolds in spheres. Ark. Mat.
51 (2013),197–221.[2] A. Balmu¸s, S. Montaldo, C. Oniciuc. Biharmonic hypersurfaces in 4-dimensional space forms.
Math.Nachr.
283 (2010), 1696–1705.[3] A. Balmu¸s, S. Montaldo, C. Oniciuc. Classification results for biharmonic submanifolds in spheres.
IsraelJ. Math.
168 (2008), 201–220.[4] A. Balmu¸s, C. Oniciuc. Biharmonic submanifolds with parallel mean curvature vector field in spheres.
J. Math. Anal. Appl.
386 (2012), 619–630.[5] A. Balmu¸s, C. Oniciuc. Biharmonic surfaces of S . Kyushu J. Math.
63 (2009), 339–345.[6] R. Caddeo, S. Montaldo, C. Oniciuc. Biharmonic submanifolds in spheres.
Israel J. Math.
130 (2002),109–123.[7] R. Caddeo, S. Montaldo, C. Oniciuc. Biharmonic submanifolds of S . Internat. J. Math.
12 (2001),867–876.[8] B.-Y. Chen, Some open problems and conjectures on submanifolds of finite type,
Soochow J. Math.
Riemannian Geometry . Birkh¨auser, 1992.[10] J. Eells, L. Lemaire.
Selected topics in harmonic maps.
CBMS Regional Conference Series in Mathe-matics, 50. American Mathematical Society, Providence, RI, 1983.[11] G.Y. Jiang. 2-harmonic maps and their first and second variation formulas.
Chinese Ann. Math. Ser. A7 , 7 (1986), 130–144.[12] S. Montaldo, C. Oniciuc. A short survey on biharmonic maps between riemannian manifolds.
Rev. Un.Mat. Argentina , 47 (2006), 1–22.[13] S. Montaldo, A. Ratto. Biharmonic curves into quadrics, arXiv:1309.0631.[14] Y.-L. Ou, L. Tang. On the generalized Chen’s conjecture on biharmonic submanifolds.
Michigan Math.J. , 61 (2012), 531–542.[15] C. Oniciuc. Biharmonic maps between Riemannian manifolds. An. Stiint. Univ. Al.I. Cuza Iasi Mat(N.S.) 48 (2002), 237–248.
Universit`a degli Studi di Cagliari, Dipartimento di Matematica e Informatica, Via Ospedale72, 09124 Cagliari, Italia
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E-mail address : [email protected]@unica.it