Biharmonic maps betweenRiemannian manifolds

R. Caddeo1, E. Loubeau2, S. Montaldo1, C. Oniciuc3, P. Piu1

1Department of Mathematics, Cagliari, Italy - 2Department of Mathematics, Brest, France - 3Faculty of Mathematics, Iasi, Romania

caddeo@unica.it - loubeau@univ-brest.fr - montaldo@unica.it - oniciucc@uaic.ro - piu@unica.it

2000 Mathematics Subject Classification. 58E20

Poster N. 57 - Scientific Section N. 5

The definition

Let ϕ : (M, g) → (N, h) be a smooth map between Rieman-nian manifolds

Harmonic maps are critical points of the energy functional

E(ϕ) =1

2

∫

M|dϕ|2vg

The corresponding Euler-Lagrange equation is ([8])

τ (ϕ) = traceg∇dϕ = 0

Biharmonic maps are the critical points of the bienergy func-tional

E2(ϕ) =1

2

∫

M|τ (ϕ)|2 vg,

and are solutions of the following Euler-Lagrange equationassociated to E2 ([10])

τ2(ϕ) = −∆τ (ϕ)− traceg RN (dϕ, τ (ϕ))dϕ = 0

∆ = − traceg(

∇ϕ∇ϕ −∇ϕ∇

)

is the rough LaplacianRN (X,Y ) = [∇X ,∇Y ]−∇[X,Y ] is the curvature on N

General properties

• A harmonic map is obviously a biharmonic map and anabsolute minimum of the bienergy. A non-harmonic bi-harmonic map is called proper biharmonic.

•E2 does not satisfy Palais-Smale condition (C) ifdim(M ) ≤ 3.

A proper biharmonic map does not exist if:

•M is compact and RiemN ≤ 0 ([10])

• ϕ : M → N is an isometric immersion with |τ (ϕ)| =constant and RiemN ≤ 0 ([14])

• ϕ : M2 → N3 is an isometric immersion and N 3 has con-stant non positive curvature ([7, 4])

• ϕ is a Riemannian submersion with basic tension fieldand one of the following holds ([14]):

a) M is compact, orientable and RicciN ≤ 0

b) RicciN < 0 and |τ (ϕ)| is constantc) N is compact and RicciN < 0

Generalized Chen’s Conjecture: Biharmonic submani-folds of a manifold N with RiemN ≤ 0 are minimal

Main Examples and Constructions

The following maps are proper biharmonic:

• Any polynomial map of degree 3 between Euclideanspaces

• The generalized Kelvin transformation ([1])

ϕ : Rm \ {0} → R

m \ {0}, ϕ(p) = p|p|m−2 , m ≥ 4

• The standard immersion of the generalized Clifford torusinto the sphere [10]

i : Sp( 1√

2)× S

q( 1√2) → S

n+1(1), p 6= q, p + q = n

The standard immersion of the parallel hyper-sphere ([4]) S

n( 1√2) → S

n+1(1)

Property: The composition

Mϕ−→ S

n( 1√2)

i−→ Sn+1(1) (1)

is proper biharmonic if and only if ϕ is harmonic with con-stant energy density e(ϕ) = 1

2|dϕ|2 ([12, 4]).

Application: There exist closed orientable embeddedproper biharmonic surfaces of arbitrary genus in S

4 ([4]).

Classification results

Proper biharmonic curves on a surface of revolution are classified, for example we have ([2]):

Biharmonic curves on a ondu-

loid

Biharmonic curves on a

nodoid

Biharmonic curves on a torus

A biharmonic curve on a

sphere (in red) tangent to a

harmonic curve (in green)

Proper biharmonic curves on the eight 3-dimensional Thurston geometries are classified bymeans of Cartan-Vranceanu metrics ds2

`,m = dx2+dy2

[1+m(x2+y2)]2+

(

dz + `2

ydx−xdy[1+m(x2+y2)]

)2

, `,m ∈ R. They are:

helices (curves with con-stant geodesic curvatureand geodesic torsion) forany value of m and ` andtheir parametrization isdescribed explicitly in ([6]).In particular:

If `2 = 4m, then (M, ds2`,m) is the

standard 3-sphere and the bihar-

monic helices are ([3]):

- circles of radius 1√2

- geodesics of the Clifford torus

S1( 1√

2)× S

1( 1√2) ⊂ S

3

If m = 0, then (M,ds2`,m)

is the Heisenberg space

and the biharmonic he-

lices are the intersec-

tion of a cylinder with a

”helicoid” and they are

geodesics of the cylinder

([5]).

Surfaces• A proper biharmonic surface of S

3 is locally a piece of S2( 1√

2) ⊂ S

3. If compact, it is S2( 1√

2)

([3]).

• A Hopf cylinder Sγ ⊂ N3(c), c > 1, in a Sasakian space form is proper biharmonic if γ is aRiemannian circle ([9]).

The stress-energy tensor

As described by Hilbert, the stress-energy tensor associated to a variational problem is a sym-metric 2-covariant tensor field S conservative at critical points, i.e. div S = 0 at these points.

In the context of harmonic maps, the stress-energy tensor is

S =1

2|dϕ|2g − ϕ∗h, with div S = −〈τ (ϕ), dϕ〉

For biharmonic maps the stress-energy tensor is ([11, 13])

S2(X,Y ) =1

2|τ (ϕ)|2〈X,Y 〉 + 〈dϕ,∇τ (ϕ)〉〈X,Y 〉−〈dϕ(X),∇Y τ (ϕ)〉 − 〈dϕ(Y ),∇Xτ (ϕ)〉, with div S2 = −〈τ2(ϕ), dϕ〉

Property: The tensor S (S2) vanishes precisely at critical points of the energy (bienergy) forvariations of the domain metric, rather than for variations of the map ([16, 13]).

A map ϕ : (M, g) → (N, h) with S2 = 0 is harmonic if([11, 13]):

• dim(M ) = 2

•M is compact and orientable with dim(M ) 6= 4

• ϕ is an isometric immersion and dim(M ) 6= 4

•M is complete and ϕ has finite energy and bienergy

•M is compact, dim(M ) = dim(N ) = 4 and rank ϕ = 4

Two characterizations ([11, 13]):

• An isometric immersionϕ : (M4, g) → (N, h) has S2 = 0 ifand only if it is pseudo-umbilical

• A hypersurface i : Mm → Nm+1,m 6= 4, has ∇S2 = 0 if and only if itis parallel

Feature: S2 is useful in finding new biharmonic maps.

Application: A submersion ϕ : (M, g) → (N, h) with basic tension field τ (ϕ) = ζ ◦ϕ is biharmonicif ζ is Killing and of constant norm ([13]).

Stability

Let ϕ : (M, g) → Sn be a biharmonic map. Then the Hessian

of the bienergy at ϕ is given by ([10, 15])

H2(V,W ) =

∫

M〈I(V ),W 〉vg,

where I(V ) is the Jacobi operator of the bienergy.

Although I has a rather complicated form ([15]), in the caseof the identity map Id of S

n it becomes

IId(V ) = ∆(∆V )− 2(n− 1)∆V + (n− 1)2V,and we deduce

a) if n = 2, then nullity(Id) = 6

b) if n > 2, then nullity(Id) =n(n+1)

2

• All the biharmonic maps constructed using (1), of Exam-ples and Constructions, are unstable.

• The biharmonic index of the canonical inclusion i :Sn−1( 1√

2) → S

n is exactly 1, and its nullity is n(n−1)2 + n

([12]).

Conjecture: The biharmonic index of Sm(1/2)×Sm(1/2) →S2m+1(1/

√2) → S2m+2 is exactly 1

Further Studies

The hyperspheres Sm( 1√

2) and the generalized Clifford torus

are the only known examples of proper biharmonic hyper-surfaces of S

m+1.

Open problem: Prove that proper biharmonic hypersur-faces of S

m+1 have constant mean curvature

Open problem: Classify all proper biharmonic hypersur-faces of S

m+1

Open problem: Compute the biharmonic index of the gen-eralized Clifford torus in S

m+1

Harmonic maps do not always exists, for instance, J. Eellsand J.C. Wood showed that there exists no harmonic mapfrom T

2 to S2 (whatever the metrics chosen) in the homotopy

class of Brouwer degree ±1.

Open problem: Find a biharmonic map from T2 to S

2

The Ruh-Vilms theorem asserts that an isometric immer-sion ϕ : M → R

n has parallel mean curvature vector field ifand only if the Gauss map γ : M → Gm(Rn) is harmonic

Open problem: Characterize isometric immersions ϕ :M → R

n with biharmonic Gauss map

References

[1] A. Balmus, S. Montaldo, C. Oniciuc. Biharmonic maps between warped product manifolds. J. Geom.Phys, to appear.

[2] R. Caddeo, S. Montaldo, P. Piu. Biharmonic curves on a surface. Rend. Mat. Appl., 21 (2001),143–157.

[3] R. Caddeo, S. Montaldo, C. Oniciuc. Biharmonic submanifolds of S3. Int. J. Math., 12 (2001), 867–

876.

[4] R. Caddeo, S. Montaldo, C. Oniciuc. Biharmonic submanifolds in spheres. Israel J. Math., 130(2002), 109–123.

[5] R. Caddeo, C. Oniciuc, P. Piu. Explicit formulas for non-geodesic biharmonic curves of the Heisen-berg group. Rend. Sem. Mat. Univ. Politec. Torino, 62 (2004), 265–278.

[6] R. Caddeo, S. Montaldo, C. Oniciuc, P. Piu. The classification of biharmonic curves of Cartan-Vranceanu 3-dimensional spaces. arXiv:math.DG/0510435.

[7] B.-Y. Chen. Some open problems and conjectures on submanifolds of finite type. Soochow J. Math.,17 (1991), 169–188.

[8] J. Eells, J.H. Sampson. Harmonic mappings of Riemannian manifolds. Amer. J. Math., 86 (1964),109–160.

[9] J. Inoguchi. Submanifolds with harmonic mean curvature in contact 3−manifolds. Colloq. Math.,100 (2004), 163–179 .

[10] G.Y. Jiang. 2-harmonic maps and their first and second variational formulas. Chinese Ann. Math.Ser. A, 7 (1986), 389–402.

[11] G.Y. Jiang. The conservation law for 2-harmonic maps between Riemannian manifolds. Acta Math.Sinica, 30 (1987), 220–225.

[12] E. Loubeau, C. Oniciuc. The index of biharmonic maps in spheres. Compos. Math. 141 (2005),729–745.

[13] E. Loubeau, S. Montaldo, C. Oniciuc. On the stress-energy tensor for biharmonic maps.arXiv:math.DG/0602021.

[14] C. Oniciuc. Biharmonic maps between Riemannian manifolds. An. Stiint. Univ. Al.I. Cuza Iasi Mat.(N.S.), 48 (2002), 237–248.

[15] C. Oniciuc. On the second variation formula for biharmonic maps to a sphere. Publ. Math. Debrecen,61 (2002), 613–622.

[16] A. Sanini. Applicazioni tra varieta riemanniane con energia critica rispetto a deformazioni di metri-che. Rend. Mat., 3 (1983), 53–63.

The bibliography of biharmonic maps

http://beltrami.sc.unica.it/biharmonic/