Some global minimizers of a symplectic Dirichlet energy
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SOME GLOBAL MINIMIZERS OF A SYMPLECTIC DIRICHLET ENERGY
J.M. SPEIGHT AND M. SVENSSON
Abstract.
The variational problem for the functional F = R M k ϕ ∗ ω k is considered, where ϕ : ( M, g ) → ( N, ω ) maps a Riemannian manifold to a symplectic manifold. This functionalarises in theoretical physics as the strong coupling limit of the Faddeev-Hopf energy, and may beregarded as a symplectic analogue of the Dirichlet energy familiar from harmonic map theory.The Hopf fibration π : S → S is known to be a locally stable critical point of F . It is provedhere that π in fact minimizes F in its homotopy class and this result is extended to the casewhere S is given the metric of the Berger’s sphere. It is proved that if ϕ ∗ ω is coclosed then ϕ is a critical point of F and minimizes F in its homotopy class. If M is a compact Riemannsurface, it is proved that every critical point of F has ϕ ∗ ω coclosed. A family of holomorphichomogeneous projections into Hermitian symmetric spaces is constructed and it is proved thatthese too minimize F in their homotopy class. Introduction
Given a smooth map φ : M → N between Riemannian manifolds, a naturalnotion of the energy of φ is the Dirichlet energy, E = 12 Z M k d φ k . The variational problem for E , whose critical points are harmonic maps, has beenheavily studied for many years. If we replace the metric on N by a symplectic form ω , a natural analogue of E is F = 12 Z M k φ ∗ ω k , which one can regard as a kind of symplectic Dirichlet energy. In this paper we studythe variational problem for F , focussing particularly on the problem of obtainingglobal minimizers of F .Our original motivation comes from theoretical physics. If N is a K¨ahler manifold,both E and F are defined, and the variational problem for E + αF , where α is apositive constant called the coupling constant, is known to physicists as the Faddeev-Hopf (or Faddeev-Skyrme) model. The case of most interest is M = R , N = S .This variational problem, originally proposed by Faddeev in the 1970s, lay dormantfor lack of computational power until the 1990s, but has been the subject of intensenumerical study in recent years [3, 8, 12, 16]. Its critical points, which are interpretedas topological solitons, have been proposed as models of gluon flux tubes in hadrons(particles composed of quarks). There has been some analytic progress on thismodel too. Kapitanski and Vakulenko proved a rather curious topological energybound for maps R → S [18], Kapitanski and Auckly proved weak existence (i.e.existence in some Sobolev space with low regularity) of global minimizers in every Mathematics Subject Classification. omotopy class of maps M → S , with M a compact oriented 3-manifold [1], andWard obtained some exact results for the case S → S [19].The variational problem for F , which we shall consider here, can be interpretedas the Faddeev-Hopf model in the strong coupling limit α → ∞ . This has two keysimilarities with Yang-Mills theory: it is conformally invariant in dimension 4 and itpossesses an infinite dimensional symmetry group (the symplectic diffeomorphismsof N ). As in Yang-Mills, the most physically relevant choice of M is S , regarded asthe conformal compactification of Euclideanized spacetime R . It is an interestingopen question whether there is a critical point of F in the generator of π ( S ).Such a critical point would be interpreted as an instanton in the strongly coupledFaddeev-Hopf model on Minkowski space.This variational problem seems to have received remarkably little attention. Fer-reira and De Carli [6] analyzed the case M = S × R , with a Lorentzian metric, and N = C , S or the hyperbolic plane, working within a particular rotationally invari-ant ansatz. The first systematic development of the variational calculus was madein [15]. We begin by briefly reviewing some results from that paper. Throughout,all maps are smooth, ( M, g ) is a Riemannian manifold and (
N, ω ) is a symplecticmanifold. In the case where N is K¨ahler, ω is the K¨ahler form ω ( · , · ) = h ( J · , · ), h isthe metric and J is the almost complex structure. The coderivative on differentialforms will be denoted δ . Theorem 1.1. [15]
For any variation ϕ t of ϕ : M → N with variational vectorfield X ∈ Γ( ϕ − T N ) we have ddt F ( ϕ t ) (cid:12)(cid:12) t =0 = Z M ω ( X, d ϕ ( ♯ δϕ ∗ ω )) ∗ . In particular, ϕ is a critical point of F if and only if d ϕ ( ♯ δϕ ∗ ω ) = 0 . Theorem 1.2. [15]
A Riemannian submersion ϕ : M → N from a Riemannianmanifold to a K¨ahler manifold is a critical point of F if and only if it has minimalfibres, i.e., if and only if it is a harmonic map. A well known harmonic Riemannian submersion is the Hopf map π : S → S , where S is the sphere in R of radius 1, and S the sphere in R of radius 1 /
2. Asa harmonic map, π is well known to be unstable , as are all harmonic maps from S [20]. We studied in [15] the second variation of F for maps into K¨ahler manifolds,and found the associated Jacobi operator. Theorem 1.3. [15]
Let N be K¨ahler and ϕ : M → N be a critical point of F . Thenthe Hessian of ϕ is H ϕ ( X, Y ) = Z M h ( X, L ϕ Y ) ∗ , where L ϕ Y = − J {∇ ϕZ ϕ Y + d ϕ [ ♯ δ d ϕ ∗ ( Y y ω )] } and Z ϕ = ♯ δϕ ∗ ω. y a careful calculation of the spectrum of this operator for the Hopf map, weproved the following result, which was conjectured by Ward [19]. Theorem 1.4. [15]
The Hopf map π : S → S is stable for the full Faddeev-Hopffunctional E + αF if and only if α ≥ . In particular, the Hopf map is a stable critical point of F . In this paper, westrengthen this result to show that the Hopf map in fact minimizes F in its homo-topy class. The same is true for the Hopf map from the Berger’s spheres π : ( S , g t ) → S , for all 0 < t ≤
1; see Example 2.2 for the definition of Berger’s spheres. It isinteresting to note that a slightly stronger version of Theorem 1.4 (namely, that π is a local minimizer of E + αF when α >
1) was obtained independently by Isobe[11] using rather different methods. It remains an open question whether the Hopfmap globally minimizes E + αF for some α .As proved in [15], we have H ϕ ( X, X ) = Z M ω ( X, ∇ ϕZ ϕ X ) ∗ k d ϕ ∗ ( X y ω ) k L ( X ∈ Γ( ϕ − T N ) . In particular, when ϕ ∗ ω is coclosed (so Z ϕ = 0), ϕ is a stable critical point of F . Inthis paper, we strengthen this by showing that, if ϕ ∗ ω is coclosed, then ϕ actuallyminimizes F in its homotopy class.2. Global minimizers
Denote by S the unit sphere in R of radius 1, and by S the sphere in R ofradius 1 / Theorem 2.1.
The Hopf map π : S → S minimizes F in its homotopy class. The proof makes use of the
Hopf invariant of a map ϕ : S → S . Recall thatthis is defined as the number H ( ϕ ) = Z S d α ∧ α, where d α = ϕ ∗ ω , and ω is the volume form of S . It is well known that this isindependent of the choice of α , and depends only on the homotopy class of ϕ , seee.g. [5]. Proof.
As above, assume that ϕ : S → S is any map and write ϕ ∗ ω = d α . By theHodge decomposition, we may assume that α is coexact. Then F ( ϕ ) = 12 h d α, d α i L = 12 h α, ∆ α i L ≥ λ k α k L , where λ is the first eigenvalue of the Hodge-Laplace operator on coexact 1-formson S . It is known that λ = 4, see e.g., [9, page 270]. Hence k α k L ≤ F ( ϕ ) . y Cauchy-Schwarz, H ( ϕ ) ≤ k ϕ ∗ ω k L k α k L ≤ √ k ϕ ∗ ω k L p F ( ϕ ) = F ( ϕ ) . For the Hopf map π : S → S , it is well known that (see e.g. [10, page 102])d ∗ π ∗ ω = 2 π ∗ ω, so that H ( π ) = F ( π ) . Thus, if ϕ and π are homotopic, then F ( π ) = H ( π ) = H ( ϕ ) ≤ F ( ϕ ) . (cid:3) Example 2.2.
Consider again the Hopf map π : S → S . We may write themetric g on S as g = g V + g H , where V is the distribution tangent to the fibres of π , and H its orthogonal comple-ment. For 0 < t <
1, the 3-dimensional
Berger’s sphere is the Riemannian manifold( S , g t ), where g t = t g V + g H . It is easy to see that d ∗ π ∗ ω = 2 tπ ∗ ω. For 0 < t <
1, a simple calcuation shows that the sectional curvature of g t isbounded below by t . Hence, the minimal eigenvalue for the Hodge-Laplace operatoron coexact 1-forms with respect to g t is bounded below by 4 t , see [9, page 270].Thus, for any map ϕ homotopic to π , a calculation similar to that above gives F ( π ) = tH ( π ) = tH ( ϕ ) ≤ F ( ϕ ) . So π still minimizes F in its homotopy class, as a map from the Berger’s sphere to S .Next we consider another class of maps which minimize F . Theorem 2.3.
Let M be compact and ϕ : M → N have ϕ ∗ ω coclosed. Then ϕ minimizes F in its homotopy class.Proof. Let ϕ : M → N have ϕ ∗ ω coclosed, ψ : M → N be homotopic to ϕ and ϕ t be a smooth homotopy from ϕ to ψ . By the homotopy lemma [7] ddt h ϕ ∗ t ω, ϕ ∗ ω i L = h d( ϕ ∗ t X y ω ) , ϕ ∗ ω i L = h ϕ ∗ t X y ω, δϕ ∗ ω i L = 0 . Hence, by Cauchy-Schwartz, F ( ϕ ) = 12 h ψ ∗ ω, ϕ ∗ ω i L ≤ p F ( ψ ) F ( ϕ ) , so that F ( ϕ ) ≤ F ( ψ ) . (cid:3) emark 2.4. Note that if δϕ ∗ ω = 0 then ϕ ∗ ω is harmonic, and thus minimizesthe L -norm in its cohomology class. If ψ is homotopic to ϕ , then ψ ∗ ω is in thiscohomology class, and this gives an alternative proof of the above result. Corollary 2.5.
Any critical immersion from a compact Riemannian manifold to asymplectic manifold minimizes F in its homotopy class. Clearly, any symplectomorphism on a compact symplectic Riemannian manifoldis a minimizer in its homotopy class. In particular, we have
Corollary 2.6.
The identity map on a compact symplectic Riemannian manifoldminimizes F in its homotopy class. We next prove that if M is a compact Riemann surface, all critical points of F have ϕ ∗ ω coclosed. Theorem 2.7.
Let M be a compact, oriented, 2-dimensional Riemannian manifoldand N be a symplectic manifold. Then every critical point ϕ : M → N of F has ϕ ∗ ω coclosed (and hence minimizes F in its homotopy class).Proof. For any ϕ : M → N , ϕ ∗ ω = f ∗ , where ∗ M and f is a real function on M . Then δϕ ∗ ω = ∗ d f ,so that ♯ δϕ ∗ ω = J ∇ f, where J is the Hermitian structure on M associated with the orientation.Assume now that ϕ is critical. Then d ϕ ( J ∇ f ) = 0 so that0 = ( ϕ ∗ ω )( J ∇ f, ∇ f ) = − f |∇ f | . It follows that ∇ f = 0 everywhere, so f is constant. (Assume, to the contrary, that( ∇ f )( p ) = 0 for some p ∈ M . Then there is some neighbourhood of p on which ∇ f = 0. But then f = 0 on this neighbourhood, so ( ∇ f )( p ) = 0, a contradiction.)Hence ϕ ∗ ω is coclosed. (cid:3) For our next result, recall that a submersion ϕ : ( M, g ) → ( N, h, J ) from aRiemannian manifold to an almost Hermitian manifold gives rise to an f -structure f on M , such that ker f = ker d ϕ and the restriction of f to (ker d ϕ ) ⊥ correspondsto J under the identification (ker d ϕ ) ⊥ ∼ = ϕ ∗ T N (see [14] for the definition of an f -structure). The map is said to be pseudo horizontally (weakly) conformal ( PHWC )if f is skew-symmetric with respect to the metric g on M . In particular, if M alsocarries an almost Hermitian structure with respect to which ϕ is holomorphic, then ϕ is PHWC . See e.g. [2, 13] for other characterizations of
PHWC maps.
Proposition 2.8.
Assume that ϕ : ( M, g ) → ( N, h, J ) is a PHWC submersionfrom a Riemannian manifold to an almost Hermitian manifold, with associated f -structure f . Then δϕ ∗ ω = f div f y ϕ ∗ ω − X a h ( ϕ ∗ f e a , ∇ d ϕ ( e a , · )) , where in the last term we sum over a local orthonormal frame for T M . roof. For any vector field Y on M we have δϕ ∗ ω ( Y ) = − X a (cid:0) e a ϕ ∗ ω ( e a , Y ) − ϕ ∗ ω ( ∇ e a e a , Y ) − ϕ ∗ ω ( e a , ∇ e a Y ) (cid:1) = − X a (cid:0) h ( ∇ e a ϕ ∗ f e a − ϕ ∗ f ∇ e a e a , ϕ ∗ Y ) + h ( ϕ ∗ f e a , ∇ d ϕ ( e a , Y )) (cid:1) = − X a h ( ∇ d ϕ ( e a , f e a ) , ϕ ∗ Y ) − h ( ϕ ∗ div f, ϕ ∗ Y ) − X a h ( ϕ ∗ f e a , ∇ d ϕ ( e a , Y ))To see that the first term is zero, we can choose the local orthonormal framesuch that e , . . . , e n span ker d ϕ = ker f , while e n +1 , . . . , e m span the orthogonalcomplement of ker d ϕ . Evaluating the sum for the two local orthonormal frames { e , . . . , e n , e n +1 , . . . , e m } and { e , . . . , e n , f e n +1 , . . . , f e m } , shows that this amountsto zero. Furthermore, the second term equals f div f y ϕ ∗ ω ( Y ), and the proof isfinished. (cid:3) Assume that ϕ : ( M, g ) → ( N, h ) is a submersion between two Riemannianmanifolds, and denote by V the vertical distribution ker d ϕ and by H the horizontal distribution V ⊥ on M . Recall that ϕ is said to be horizontally conformal if there isa positive function λ on M , the dilation of ϕ , such that h ( ϕ ∗ X, ϕ ∗ Y ) = λ g ( X, Y ) (
X, Y ∈ Γ( H )) . If λ is constant, ϕ is said to be horizontally homothetic . Clearly, any horizontallyconformal submersion to an almost Hermitian manifold is PHWC . Corollary 2.9.
Assume that ϕ : ( M, g ) → ( N, h, J ) is a horizontally homotheticsubmersion from a Riemannian manifold to an almost Hermitian manifold. Then δϕ ∗ ω = − λ ♭ div f, where λ is the dilation of ϕ and f the associated f -structure.Proof. It is well known that ∇ d ϕ ( X, Y ) = 0 when X and Y are horizontal. Thus,if Y is horizontal, we have δϕ ∗ ω ( Y ) = h ( J ϕ ∗ f div f, ϕ ∗ Y ) = − λ g (div f, Y ) . On the other hand, if Y is vertical, then δϕ ∗ ω ( Y ) = − X a h ( ϕ ∗ f e a , ∇ d ϕ ( e a , Y )) = X a h ( ϕ ∗ f e a , ϕ ∗ ∇ e a Y )= − λ X a g ( ∇ e a f e a , Y ) = − λ g (div f, Y ) . (cid:3) We will use this result to construct a family of homogeneous projections intoHermitian symmetric spaces, all of which minimize F in their homotopy class. Let g C be a semi-simple complex Lie algebra, g be a compact real form of g C and h ⊂ g be a Cartan subalgebra with complexification h C ⊂ g C . Denote by ∆ = ∆ + ∪ ∆ − the set of roots and its decomposition into positive and negative roots, after a choice f a positive Weyl chamber. Let Π ⊂ ∆ + be the set of simple roots. For any subsetΠ ⊂ Π, we can construct a parabolic subalgebra as p = h C + X α ∈ [Π ] g α + X α ∈ ∆ + \ [Π ] g α , where [Π ] is the set of roots in the span of Π .Let G C be a complex Lie group with Lie algebra g C , and G , P and K be Liesubgroups of G C with Lie algebras g , p and k = g ∩ p , respectively. Let m bethe Killing orthogonal complement to k , so that m = X α ∈ ∆ + \ [Π ] ( g α + g − α ) ∩ g . We may identify m with the tangent space of G/K at the identity coset. Now G/K has a G -invariant integrable complex structure, for which the (1 ,
0) and (0 , m , = X α ∈ ∆ + \ [Π ] g α , m , = X α ∈ ∆ + \ [Π ] g − α . It is well known that minus the Killing form of g equips G/K with a Riemannianmetric for which this complex structure is Hermitian and cosymplectic (i.e. theK¨ahler form is coclosed ), see e.g. [17].Now, for any nested pair of subsets Π ⊂ Π ′ ⊂ Π of simple roots, we get, withobvious notation, a homogeneous fibration ϕ : G/K → G/K ′ , and this map is clearly a holomorphic Riemannian submersion with totally geodesicfibres, see e.g., [4, page 257]. Hence, in the case where G/K ′ is K¨ahler, ϕ is a criticalpoint of F . From now on, we will assume that G/K ′ is a Hermitian symmetric space,so that the complex structure is in fact K¨ahler. Proposition 2.10.
With the notation and conventions introduced above, the homo-geneous fibration ϕ : G/K → G/K ′ minimizes F in its homotopy class.Proof. It is, by Theorem 2.3, Corollary 2.9 and G -invariance, enough to show thatdiv f = 0 at the identity coset. It follows easily from the formula for the Levi-Civitaconnection in a reductive homogeneous space, see [4, page 183], that h∇ X Y, Z i = 0 , for all X, Y ∈ g ± α and Z ∈ g ± β , for any α, β ∈ ∆ + . Since f is G -invariant andpreserves the subspace ( g α + g − α ) ∩ g of m , we have h ( ∇ X f ) X, Z i = h∇ X f X, Z i − h f ∇ X X, Z i = 0 , for all X ∈ g ± α and Z ∈ g ± β , for any α, β ∈ ∆ + . From the above orthogonaldecomposition of m , we see that div f = 0. (cid:3) Example 2.11.
Let n , . . . , n k be positive integers, and n = n + · · · + n k . Define M as the space of decompositions C n = V ⊕ · · · ⊕ V k , here dim V i = n i , and V i ⊥ V j whenever i = j . If we denote by Gr n ( C n ) theGrassmannian of n -dimensional subspaces of C n , we have an obvious map ϕ : M → Gr n ( C n ) , ϕ ( V ⊕ · · · ⊕ V k ) = V . By considering both M and Gr n ( C n ) as homogeneous SU ( n )-spaces, it followseasily that ϕ is a homogenous projection of the type considered above, and hence aminimizer of F in its homotopy class. Remark 2.12.
Let ϕ : M → N be a holomorphic map between almost Hermitianmanifolds. It is well known that if M and N are almost K¨ahler then ϕ is harmonicand minimizes the Dirichlet energy E in its homotopy class, see e.g., [7] for a proof.In fact, this proof immediately generalizes to the case where M is only cosymplectic.Hence, if, in addition, ϕ ∗ ω is coclosed, we see that ϕ minimizes the full Faddeev-Hopf energy E + αF for all α >
0. This applies to the homogeneous fibrations
G/K → G/K ′ defined above. References [1] D. Auckly and L. Kapitanski,
Analysis of the Faddeev Model
Commun. Math. Phys. (2005) 611–620.[2] P. Baird and J. C. Wood,
Harmonic morphisms between Riemannian manifolds , London Math. Soc. Monogr.No. , Oxford Univ. Press, 2003.[3] R. A. Battye and P. M. Sutcliffe, Solitons, links and knots , Proc. Roy. Soc. Lond. A (1999) 4305-4331.[4] A. L. Besse,
Einstein Manifolds , Ergebnisse der Matematik und ihrer Grenzbebiete (3), vol. 10, Springer-Verlag,Berlin, 1987.[5] R. Bott and L.W. Tu,
Differential Forms in Algebraic Topology , Graduate Texts in Mathematics, 82. Springer-Verlag, New York-Berlin, 1982.[6] E. De Carli and L. A. Ferreira,
A model for Hopfions on the space-time S × R J. Math. Phys. (2005)012703.[7] J. Eells and L. Lemaire, Selected topics in harmonic maps , CBMs Regional Conference Series in Mathematics,vol. 50, AMS, Providence, RI, 1983.[8] L. D. Faddeev and A. J. Niemi,
Knots and particles , Nature : (1997) 58.[9] S. Gallot and D. Meyer,
Op´erateur de courbure et Laplacien des formes diff´erentielles d’une vari´et´e Rieman-nienne , J. Math. Pures et Appl. (1975) 259–284.[10] P. B. Gilkey, J. V. Leahy, J. Park, Spectral geometry, Riemannian submersions, and the Gromov-Lawsonconjecture , Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 1999.[11] T. Isobe,
On a minimizing property of the Hopf soliton in the Faddeev-Skyrme model , Rev. Math. Phys. (2008) 765–786.[12] J. Hietarinta and P. Salo, Ground state in the Faddeev-Skyrme model , Phys. Rev.
D62 : 081701 ( ).[13] R. Slobodeanu,
A special class of holomorphic mappings and the Faddeev-Hopf model , arXiv:0802.1626[math.DG].[14] J. H. Rawnsley, f -structures, f -twistor spaces and harmonic maps , Lecture Notes in Math. (1985),85–159.[15] J. M. Speight and M. Svensson, On the Strong Coupling Limit of the Faddeev–Hopf Model , Commun. Math.Phys. (2007) 751–773.[16] P.M. Sutcliffe,
Knots in the Skyrme-Faddeev model , Proc. Roy. Soc. Lond. A (2007) 3001-3020.[17] M. Svensson,
Harmonic morphisms in Hermitian geometry , J. Reine Angew. Math. (2004) 45–68.[18] A. F. Vakulenko and L. V. Kapitansky,
Stability of solitons in S(2) in the nonlinear sigma model , Sov. Phys.Dokl. (1979) 433–434.[19] R. S. Ward, Hopf solitons on S and R , Nonlinearity (1999) 241-246.[20] Y. L. Xin, Some results on stable harmonic maps , Duke Math. J. (1980) 609–613. School of Mathematics, University of Leeds, Leeds, LS2 9JT
E-mail address : [email protected] Department of Mathematics and Computer Science, and CP -Origins, Centre of Excellence forParticle Physics Phenomenology, University of Southern Denmark, Campusvej 55, DK-5230 Odense M E-mail address : [email protected]@imada.sdu.dk