On interpolating sesqui-harmonic maps between Riemannian manifolds
aa r X i v : . [ m a t h . DG ] J a n ON INTERPOLATING SESQUI-HARMONIC MAPS BETWEENRIEMANNIAN MANIFOLDS
VOLKER BRANDING
Abstract.
Motivated from the action functional for bosonic strings with extrinsic curvatureterm we introduce an action functional for maps between Riemannian manifolds that interpo-lates between the actions for harmonic and biharmonic maps. Critical points of this functionalwill be called interpolating sesqui-harmonic maps. In this article we initiate a rigorous mathe-matical treatment of this functional and study various basic aspects of its critical points. Introduction and Results
Harmonic maps play an important role in geometry, analysis and physics. On the one handthey are one of the most studied variational problems in geometric analysis, on the other handthey naturally appear in various branches of theoretical physics, for example as critical pointsof the nonlinear sigma model or in the theory of elasticity. Mathematically, they are defined ascritical points of the Dirichlet energy E ( φ ) = Z M | dφ | dV, (1.1)where φ : M → N is a map between the two Riemannian manifolds ( M, h ) and (
N, g ). Thecritical points of (1.1) are characterized by the vanishing of the so-called tension field , which isgiven by 0 = τ ( φ ) := Tr h ∇ dφ. This is a semilinear, elliptic second order partial differential equation, for which many resultson existence and qualitative behavior of its solutions have been obtained. For a recent surveyon harmonic maps see [15]. Due to their nonlinear nature harmonic maps do not always needto exist. For example, if M = T and N = S , there does not exist a harmonic map withdeg φ = ± biharmonic maps . These arise as critical points of the bienergy [16], which is definedas E ( φ ) = Z M | τ ( φ ) | dV. (1.2)In contrast to the harmonic map equation, the biharmonic map equation is an elliptic equationof fourth order and is characterized by the vanishing of the bitension field τ ( φ ) := ∆ τ ( φ ) − R N ( dφ ( e α ) , τ ( φ )) dφ ( e α ) , where ∆ is the connection Laplacian on φ ∗ T N , e α an orthonormal basis of T M and R N de-notes the curvature tensor of the target manifold N . We make use of the Einstein summationconvention, meaning that we sum over repeated indices.In the literature that studies analytical aspects of biharmonic maps one refers to (1.2) as theenergy functional for intrinsic biharmonic maps .For a survey on biharmonic maps between Riemannian manifolds we refer to [10] and [26]. Date : January 17, 2019.2010
Mathematics Subject Classification.
Key words and phrases. interpolating sesqui-harmonic maps; harmonic maps; biharmonic maps; bosonic stringwith extrinsic curvature term.
In this article we want to focus on the study of an action functional that interpolates betweenthe actions for harmonic and biharmonic maps E δ ,δ ( φ ) = δ Z M | dφ | dV + δ Z M | τ ( φ ) | dV (1.3)with δ , δ ∈ R .This functional appears at several places in the physics literature. In string theory it is knownas bosonic string with extrinsic curvature term , see [17, 28].On the mathematical side there have been several articles dealing with some particular aspectof (1.3). Up to the best knowledge of the author the first place where the functional (1.3) wasmentioned is [13, pp.134-135] with δ = 1 and δ >
0. In that reference it is already shownthat if the domain has dimension 2 or 3 and the target N negative sectional curvature thenthe critical points of (1.3) reduce to harmonic maps. Later it was shown in [20, p.191] that nocritical points exist if one does not impose the curvature condition on N and also assumes thatdeg φ = 1. Some analytic questions related to critical points of (1.3) have been discussed in [19]assuming δ = 2 , δ = 1. For the sake of completeness we want to mention that the functional(1.3) with δ > δ = is also presented in the survey article “A report on harmonicmaps”, see [11, p.28, Example (6.30)].In [21] the authors initiate an extensive study of (1.3) assuming δ = 1 and δ ∈ R under thecondition that φ is an immersion. They consider variations of (1.3) that are normal to theimage φ ( M ) ⊂ N . In this setup they call critical points of (1.3) biminimal immersions . Theyalso point out possible applications of their model to the theory of elasticity.Up to now there exist several results on biminimal immersions, see for example [9] for biminimalhypersurfaces into spheres, [23] for biminimal submanifolds in manifolds of non-positive cur-vature and [24] for biminimal submanifolds of Euclidean space. Instead of investigating mapsthat are immersions, we here want to put the focus on arbitrary maps between Riemannianmanifolds.The critical points of (1.3) will be referred to as interpolating sesqui-harmonic maps and aregiven by δ ∆ τ ( φ ) = δ R N ( dφ ( e α ) , τ ( φ )) dφ ( e α ) + δ τ ( φ ) , (1.4)where τ ( φ ) denotes the tension field of the map φ and by ∆ we are representing the connectionLaplacian on the vector bundle φ ∗ T N .As in the case of biharmonic maps it is obvious that harmonic maps solve (1.4). For this reasonwe are mostly interested in solutions of (1.4) that are not harmonic maps. However, we canexpect that as in the case of biharmonic maps there may be many situations in which solutionsof (1.4) will be harmonic maps. In particular, we can expect that this is the case if N hasnegative sectional curvature and δ δ >
0. This question will be dealt with in section 4. On theother hand, if δ and δ have opposite sign we might expect a different behavior of solutions of(1.4) since in this case the two terms in the energy functional (1.3) are competing with eachother and the energy functional can become unbounded from above and below.This article is organized as follows: In section 2 we study basic features of interpolating sesqui-harmonic maps. Afterwards, in section 3, we derive several explicit solutions of the interpolatingsesqui-harmonic map equation and in the last section we provide several results that characterizethe qualitative behavior of interpolating sesqui-harmonic maps.Throughout this paper we will make use of the following conventions. Whenever choosinglocal coordinates we will use Greek letters to denote indices on the domain M and Latinletters for indices on the target N . We will choose the following convention for the curva-ture tensor R ( X, Y ) Z := [ ∇ X , ∇ Y ] Z − ∇ [ X,Y ] Z such that the sectional curvature is given by K ( X, Y ) = R ( X, Y, Y, X ). For the Laplacian acting on functions f ∈ C ∞ ( M ) we choosethe convention ∆ f = div grad f , for sections in the vector bundle φ ∗ T N we make the choice∆ φ ∗ T N = Tr( ∇ φ ∗ T N ∇ φ ∗ T N ). Note that the connection Laplacian on φ ∗ T N is defined by∆ := ∇ e α ∇ e α − ∇ ∇ eα e α . N INTERPOLATING SESQUI-HARMONIC MAPS BETWEEN RIEMANNIAN MANIFOLDS 3 Interpolating sesqui-harmonic maps
In this section we analyze the basic features of the action functional (1.3) and start by calculatingits critical points.
Proposition 2.1.
The critical points of (1.3) are given by δ ∆ τ ( φ ) = δ R N ( dφ ( e α ) , τ ( φ )) dφ ( e α ) + δ τ ( φ ) , where τ ( φ ) := Tr h ∇ dφ is the tension field of the map φ .Proof. We choose Riemannian normal coordinates that satisfy ∇ ∂ t e α = 0 at the respective point.Consider a variation of the map φ , that is φ t : ( − ε, ε ) × M → N , which satisfies ∇ ∂t φ (cid:12)(cid:12) t =0 = η . Itis well-known that ddt (cid:12)(cid:12) t =0 Z M | dφ t | dV = − Z M h η, τ ( φ ) i dV. In addition, we find ddt (cid:12)(cid:12) t =0 Z M | τ ( φ t ) | dV =2 Z M h ∇ ∂t ∇ e α dφ t ( e α ) , τ ( φ t ) i dV (cid:12)(cid:12) t =0 =2 Z M ( h R N ( dφ t ( ∂ t ) , dφ t ( e α )) dφ t ( e α ) , τ ( φ t ) i + h∇ e α ∇ e α dφ t ( ∂ t ) , τ ( φ t ) i ) dV (cid:12)(cid:12) t =0 =2 Z M h η, ∆ τ ( φ ) − R N ( dφ ( e α ) , τ ( φ )) dφ ( e α ) i dV. Adding up both contributions yields the claim. (cid:3)
Solutions of (1.4) will be called interpolating sesqui-harmonic maps . Remark 2.2.
Choosing δ = 2 , δ = 1 and M = S , N = S k solutions of (1.4) were called quasi-biharmonic maps in [30]. These arise when considering a sequence of weakly intrinsicbiharmonic maps in dimension four. When taking the limit, one finds that quasi-biharmonicspheres separate at finitely many points as in many conformally invariant variational problems. Remark 2.3.
There is another way how we can think of (1.4). If we interpret the biharmonicmap equation as acting with the Jacobi-field operator J on the tension field τ ( φ ), then we mayrewrite the equation for interpolating sesqui-harmonic maps as J ( τ ( φ )) = δ δ τ ( φ ) . Since the Jacobi-field operator is elliptic it has a discrete spectrum whenever M is a closedmanifold. In this case the equation for interpolating sesqui-harmonic maps can be thought ofas an eigenvalue equation for the Jacobi-field operator. Remark 2.4.
In order to highlight the dependence of the action functional (1.3) on the metricon the domain M we write E δ ,δ ( φ, h ) = δ Z M | dφ | h dV h + δ Z M | τ h ( φ ) | h dV h , where dV h represents the volume element of the metric h . If we perform a rescaling of themetric by a constant factor ˜ h := λ h , the action functional transforms as E δ ,δ ( φ, ˜ h ) = δ Z M | dφ | h λ n − dV h + δ Z M | τ h ( φ ) | h λ n − dV h . This clearly reflects the fact that the action functional for harmonic maps is scale-invariant intwo dimensions whereas the action functional for biharmonic maps is scale-invariant in fourdimensions. We can conclude that the action for interpolating sesqui-harmonic maps is not Finding an appropriate name for solutions of (1.4) turned out to be subtle. The author would like to thankJohn Wood for suggesting the word “sesqui”.
VOLKER BRANDING scale-invariant in any dimension and we may expect that interpolating sesqui-harmonic mapsmay be most interesting if dim M = 2 , , Proposition 2.5.
For φ : M → S n ⊂ R n +1 with the constant curvature metric, the equationfor interpolating sesqui-harmonic maps (1.4) acquires the form δ (∆ φ + ( | ∆ φ | + ∆ | dφ | + 2 h dφ, ∇ ∆ φ i + 2 | dφ | ) φ + 2 ∇ ( | dφ | dφ )) = δ (∆ φ + | dφ | φ ) . (2.1) Proof.
Recall that for a spherical target the tension field has the simple form τ ( φ ) = ∆ φ + | dφ | φ. Since we assume that N = S n with constant curvature the term on the right hand side of (1.4)acquires the form − R N ( dφ ( e α ) , τ ( φ )) dφ ( e α ) = | dφ | τ ( φ ) − h dφ ( e α ) , τ ( φ ) i dφ ( e α )= | dφ | ∆ φ + | dφ | φ − h dφ ( e α ) , ∆ φ i dφ ( e α ) . Using the special structure of the Levi-Civita connection on S n ⊂ R n +1 we calculate∆ τ ( φ ) = ∇∇ (∆ φ + | dφ | φ )= ∇ ( ∇ ∆ φ + h dφ, ∆ φ i φ + ∇ ( | dφ | φ ))=∆ φ + h dφ, ∇ ∆ φ i φ + ∇ ( h dφ, ∆ φ i φ ) + ∆( | dφ | φ ) + h dφ, ∇ ( | dφ | φ ) i φ =∆ φ + | ∆ φ | φ + 2 h dφ, ∇ ∆ φ i φ + h dφ, ∆ φ i dφ + ∆( | dφ | φ ) + | dφ | φ. Combining both equations yields the claim. (cid:3)
By varying (1.3) with respect to the domain metric we obtain the energy-momentum tensor.Since the energy-momentum tensor for both harmonic and biharmonic maps is well-known inthe literature we can directly give the desired result.
Proposition 2.6.
The energy-momentum tensor associated to (1.3) is given by T ( X, Y ) = δ ( h dφ ( X ) , dφ ( Y ) i − | dφ | h ( X, Y )) (2.2)+ δ (cid:0) | τ ( φ ) | h ( X, Y ) + h dφ, ∇ τ ( φ ) i h ( X, Y ) − h dφ ( X ) , ∇ Y τ ( φ ) i − h dφ ( Y ) , ∇ X τ ( φ ) i (cid:1) , where X, Y are vector fields on M .Proof. We consider a variation of the metric on M , that is ddt (cid:12)(cid:12) t =0 h t = k, where k is a symmetric (2 , ddt (cid:12)(cid:12) t =0 Z M | dφ | dV h t = Z M k αβ (cid:0) h dφ ( e α ) , dφ ( e β ) i − | dφ | h αβ (cid:1) dV h . Deriving the energy-momentum tensor for biharmonic maps is more involved as one has to varythe connection of the domain since it depends on the metric. The energy-momentum tensorfor biharmonic maps was already presented in [16], a rigorous derivation was obtained in [22,Theorem 2.4], that is ddt (cid:12)(cid:12) t =0 Z M | τ ( φ ) | dV h t = − Z M k αβ ( 12 | τ ( φ ) | h αβ + h dφ, ∇ τ ( φ ) i h αβ − h dφ ( e α ) , ∇ e β τ ( φ ) i − h dφ ( e β ) , ∇ e α τ ( φ ) i ) dV h . Combining both formulas concludes the proof. (cid:3)
N INTERPOLATING SESQUI-HARMONIC MAPS BETWEEN RIEMANNIAN MANIFOLDS 5
It can be directly seen that the energy-momentum tensor (2.2) is symmetric. For the sake ofcompleteness we prove the following
Proposition 2.7.
The energy-momentum tensor (2.2) is divergence-free.Proof.
We choose a local orthonormal basis e α and set T αβ := T ( e α , e β ) = δ ( h dφ ( e α ) , dφ ( e β ) i − | dφ | h αβ )+ δ (cid:0) | τ ( φ ) | h αβ + h dφ, ∇ τ ( φ ) i h αβ − h dφ ( e α ) , ∇ e β τ ( φ ) i − h dφ ( e β ) , ∇ e α τ ( φ ) i (cid:1) . By a direct calculation we find ∇ e α ( h dφ ( e α ) , dφ ( e β ) i − | dφ | h αβ ) = h τ ( φ ) , dφ ( e β ) i and also ∇ e α ( 12 | τ ( φ ) | h αβ + h dφ, ∇ τ ( φ ) i h αβ − h dφ ( e α ) , ∇ e β τ ( φ ) i − h dφ ( e β ) , ∇ e α τ ( φ ) i )= h∇ e β τ ( φ ) , τ ( φ ) i + h∇ e β dφ ( e γ ) , ∇ e γ τ ( φ ) i + h dφ ( e γ ) , ∇ e β ∇ e γ τ ( φ ) i− h τ ( φ ) , ∇ e β τ ( φ ) i − h dφ ( e α ) , ∇ e α ∇ e β τ ( φ ) i − h∇ e α dφ ( e β ) , ∇ e α τ ( φ ) i − h dφ ( e β ) , ∇∇ τ ( φ ) i = h dφ ( e α ) , R N ( dφ ( e β ) , dφ ( e α )) τ ( φ ) i − h dφ ( e β ) , ∆ τ ( φ ) i , where we used the torsion-freeness of the Levi-Civita connection. Adding up both equationsyields the claim. (cid:3) Conservation laws for targets with symmetries.
In this subsection we discuss howto obtain a conservation law for solutions of the interpolating sesqui-harmonic map equationin the case that the target manifold has a certain amount of symmetry, more precisely if itpossesses Killing vector fields. A similar discussion has been performed in [4].To this end let ξ be a diffeomorphism that generates a one-parameter family of vector fields X .Then we know that ddt (cid:12)(cid:12) t =0 ξ ∗ g = L X g, where L denotes the Lie-derivative acting on the metric. In terms of local coordinates theLie-derivative of the metric is given by L X g ij = ∇ i X j + ∇ j X i . This enables us to give the following
Definition 2.8.
Let ξ be a diffeomorphism that generates a one-parameter family of vectorfields X on N . We say that X generates a symmetry for the action E δ ,δ ( φ, ξ ∗ g ) if ddt (cid:12)(cid:12) t =0 E δ ,δ ( φ, ξ ∗ g ) = Z M L X ( δ | dφ | + δ | τ ( φ ) | ) dV = 0 , where the Lie-derivative is acting on the metric g .Note that if X generates an isometry then L X g = 0 such that we have to require the existenceof Killing vector fields on the target.In the following we will make use of the following facts: Lemma 2.9. If X is a Killing vector field on the target N , then ∇ Y,Z X = − R N ( X, Y ) Z, (2.3) L X Γ kij = ∇ i ∇ j X k − R kijl X l , (2.4) where Γ kij are the Christoffel symbols on N , R kijl the components of the Riemannian curvaturetensor on N and Y, Z vector fields on N . VOLKER BRANDING
Lemma 2.10.
Let φ : M → N be a smooth solution of (1.4) and assume that N admits aKilling vector field X . Then the Lie-derivative acting on the metric g of the energy density isgiven by L X ( δ | dφ | + δ | τ ( φ ) | ) =2 δ ∇ e α h dφ ( e α ) , X ( φ ) i + 2 δ ∇ e α ( h τ ( φ ) , ∇ e α X ( φ ) i − δ h∇ e α τ ( φ ) , X ( φ ) i ) . Proof.
We choose Riemannian normal coordinates x α on M and calculate L X | dφ | =( L X g ) ij ∂φ i ∂x α ∂φ j ∂x β h αβ =2 ∇ i X j ∂φ i ∂x α ∂φ j ∂x β h αβ =2 h dφ ( e α ) , ∇ e α ( X ( φ )) i =2 ∇ e α h dφ ( e α ) , X ( φ ) i − h τ ( φ ) , X ( φ ) i . To calculate the variation of the tension field with respect to the target metric we first of allnote that L X | τ ( φ ) | =2 τ i ( φ ) τ j ( φ ) ∇ i X j + 2 g ij τ i ( φ ) L X τ j ( φ ) . Making use of the local expression of the tension field this yields L X τ j ( φ ) = L X (∆ φ j + h αβ ∂φ k ∂x α ∂φ l ∂x β Γ jkl )= h αβ ∂φ k ∂x α ∂φ l ∂x β L X Γ jkl = h αβ ∂φ k ∂x α ∂φ l ∂x β ( ∇ k ∇ l X j − R jklr X r ) , where we used (2.4) in the last step.This allows us to infer g ij τ i ( φ ) L X τ j ( φ ) = h τ ( φ ) , ∇ dφ ( e α ) ∇ e α X ( φ ) i + h R N ( τ ( φ ) , dφ ( e α )) dφ ( e α ) , X i . Combining both equations we find L X | τ ( φ ) | =2 ∇ e α h τ ( φ ) , ∇ e α X ( φ ) i − ∇ e α h∇ e α τ ( φ ) , X ( φ ) i + 2 h ∆ τ ( φ ) , X ( φ ) i + 2 h R N ( τ ( φ ) , dφ ( e α )) dφ ( e α ) , X i . The result follows by adding up both contributions. (cid:3)
Proposition 2.11.
Let φ : M → N be a smooth solution of (1.4) and assume that N admits aKilling vector field X . Then the following vector field is divergence free J α := δ h dφ ( e α ) , X ( φ ) i + δ h τ ( φ ) , ∇ e α X ( φ ) i − δ h∇ e α τ ( φ ) , X ( φ ) i . (2.5) Proof.
A direct calculation yields ∇ e α J α = δ h τ ( φ ) , X ( φ ) i + δ h dφ ( e α ) , ∇ e α X ( φ ) i | {z } =0 + δ h τ ( φ ) , ∇ e α ,e α X ( φ ) i − δ h ∆ τ ( φ ) , X ( φ ) i = h X ( φ ) , δ τ ( φ ) + δ R N ( dφ ( e α ) , τ ( φ )) dφ ( e α ) − δ ∆ τ ( φ ) i =0 , where we used (2.3) and that φ is a solution of (1.4) in the last step. (cid:3) Remark 2.12.
In the physics literature the vector field (2.5) is usually called
Noether current . N INTERPOLATING SESQUI-HARMONIC MAPS BETWEEN RIEMANNIAN MANIFOLDS 7 Explicit solutions of the interpolating sesqui-harmonic map equation
In this section we want to derive several explicit solutions to the Euler-Lagrange equation (1.4).We can confirm that solutions may have a different behavior than biharmonic or harmonicmaps.Let us start in the most simple setup possible.
Example 3.1.
Suppose that M = N = S and by s we denote the global coordinate on S .Then (1.4) acquires the form δ φ ′′′′ ( s ) = δ φ ′′ ( s ) . Taking an ansatz of the form φ ( s ) = X k a k e iks we obtain X k a k k ( δ k − δ ) e iks = 0 . Consequently, we have to impose the condition k = δ δ . In particular, this shows that theredoes not exist a solution of (1.4) on S if δ and δ have opposite sign. Example 3.2.
Consider the case M = R and N = R . In this case being interpolating sesqui-harmonic means to find a function f : M → N that solves δ ( ∂ x + ∂ y ) f = δ ( ∂ x + ∂ y ) f, where x, y denote the canonical coordinates in R . If we make a separation ansatz of the form f ( x, y ) = e αx e βy then we are lead to the following algebraic expression δ ( α + β ) = δ . Let us distinguish the following cases(1) δ = 0, that is f is biharmonic. In this case α + β = 0 and we have to choose α, β ∈ C .(2) δ = 0, that is f is harmonic. In this case there are no restrictions on α, β .(3) If δ δ > α + β > α, β ∈ R .(4) If δ δ < α + β < α, β ∈ C .This again shows that interpolating sesqui-harmonic functions may be very different from bothharmonic and biharmonic functions.3.1. Interpolating sesqui-harmonic functions in flat space.
In this section we studyinterpolating sesqui-harmonic functions in flat space.First, suppose that M = N = R and we denote the global coordinate on R by x . Then (1.4)acquires the form δ φ ′′′′ ( x ) = δ φ ′′ ( x ) . This can be integrated as φ ( x ) = δ δ ( c e q δ δ x + c e − q δ δ x ) + c x + c , where c i , i = 1 , . . . δ → δ → M = ( R n , δ ) and N = ( R , δ ) both with the Euclidean metric the equation forinterpolating sesqui-harmonic maps turns into δ ∆∆ f = δ ∆ f, (3.1) VOLKER BRANDING where f : R n → R . Although this equation is linear we may expect some analytical difficultiessince we do not have a maximum principle available for fourth order equations.A full in detail analysis of this equation is far beyond the scope of this article. Nevertheless,we will again see that solutions of (3.1) may be very different from harmonic and biharmonicfunctions. We will be looking for radial solutions of (3.1), where r := p x + . . . + x n . In thiscase the Laplacian has the form ∆ = d dr + n − r ddr . Recall that the fundamental solution of the Laplace equation in R n is given by H ∆ ( x, y ) = ( | x − y | − n , n ≥ , log | x − y | , n = 2 , whereas for biharmonic functions it acquires the form H ∆ ( x, y ) = | x − y | − n , n ≥ , log | x − y | , n = 4 , | x − y | , n = 3 . Note that we did not write down any normalization of the fundamental solutions.We cannot expect to find a unique solution to (3.1) since we can always add a harmonic functiononce we have constructed a solution to (3.1). Since we are considering R n instead of a curvedmanifold at the moment all curvature terms in (1.4) vanish and we are dealing with a linearproblem.Instead of trying to directly solve (3.1) we rewrite the equation as follows∆(∆ f − δ δ f ) = 0 . Assume that n ≥ f ( r ) − δ δ f ( r ) = r − n , which then provides an interpolating sesqui-harmonic function. This yields the following ordi-nary differential equation f ′′ ( r ) + n − r f ′ ( r ) − δ δ f ( r ) = r − n . (3.2)This equation can be solved explicitly in terms of a linear combination of Bessel functions inany dimension. Since the general solution is rather lengthy, we only give some explicit solutionsfor a fixed dimension. • Suppose that n = 3 then the solution of (3.2) is given by f ( r ) = c e − q δ δ r r + c e q δ δ r p δ /δ r − δ δ r . As in the one-dimensional case both limits δ → δ → • Suppose that n = 4 and that δ δ >
0. Then the solution of (3.2) is given by f ( r ) = c J ( q δ δ r ) r + c Y ( q δ δ r ) r + π p δ /δ r ( J ( r δ δ r ) Y ( r δ δ r ) − J ( r δ δ r ) Y ( r δ δ r )) . If δ δ < • The qualitative behavior of solutions to (3.2) for n ≥ N INTERPOLATING SESQUI-HARMONIC MAPS BETWEEN RIEMANNIAN MANIFOLDS 9
It becomes obvious that solutions of (3.1) may show a different qualitative behavior comparedto biharmonic functions. Moreover, as one should expect, the qualitative behavior dependsheavily on the sign of the product δ δ .3.2. Interpolating sesqui-harmonic curves on the three-dimensional sphere.
In thissubsection we study interpolating sesqui-harmonic curves on three-dimensional spheres with theround metric, where we follow the ideas from [7].To this end let (
N, g ) be a three-dimensional Riemannian manifold with constant sectionalcurvature K . Moreover, let γ : I → N be a smooth curve that is parametrized by arc length.Let { T, N, B } be an orthonormal frame field of T N along the curve γ . Here, T = γ ′ is theunit tangent vector of γ , N the unit normal field and B is chosen such that { T, N, B } forms apositive oriented basis.In this setup we have the following Frenet equations for the curve γ ∇ T T = k g N, ∇ T N = − k g T + τ g B, ∇ T B = − τ g N. (3.3) Lemma 3.3.
Let γ : I → N be a curve in a three-dimensional Riemannian manifold. Then thecurve γ is interpolating sesqui-harmonic if the following equation holds ( − δ k g k ′ g ) T + ( δ ( k ′′ g − k g − k g τ g + k g K ) − δ k g ) N + δ (2 k ′ g τ g + k g τ ′ g ) B = 0 . Proof.
Making use of the Frenet equations (3.3) a direct calculation yields ∇ T T = ( − k g k ′ g ) T + ( k ′′ g − k g − k g τ g ) N + (2 k ′ g τ g + k g τ ′ g ) B = 0 . Using that the sectional curvature of N is given by K = R N ( T, N, N, T ) we obtain the claim. (cid:3)
Corollary 3.4.
Let γ : I → N be a curve in a three-dimensional Riemannian manifold. Thenthe curve γ is interpolating sesqui-harmonic if the following system holds k g k ′ g = 0 , k ′ g τ g + k g τ ′ g = 0 , δ ( k ′′ g − k g − k g τ g + k g K ) = δ k g . The non-geodesic solutions ( k g = 0) of this system are given by k g = const = 0 , τ g = const, δ ( k g + τ g ) = δ K − δ . (3.4)We directly obtain the following characterization of interpolating sesqui-harmonic curves: Proposition 3.5. (1)
Let γ : I → N be a curve in a three-dimensional Riemannian mani-fold. If K ≤ δ δ then any interpolating sesqui-harmonic curve is a geodesic. (2) To obtain a non-geodesic interpolating sesqui-harmonic curve γ : I → S we have todemand that δ > δ . Proposition 3.6.
Let γ : I → S be a curve on the three-dimensional sphere with the roundmetric. The curve γ is interpolating sesqui-harmonic if the following equation holds γ ′′′′ + (1 − δ + δ ) γ ′′ + ( − k g − δ + δ ) γ = 0 . (3.5) Proof.
Differentiating the first equation of (3.3) we find ∇ T N = − k ′ g T − k g ∇ T T + τ ′ g B + τ g ∇ T B = − k g ∇ T T + τ g ∇ T B = − ( k g + τ g ) N =( − δ + δ ) N, where we used (3.4). Moreover, employing the formula for the Levi-Civita connection on S ⊂ R ∇ T X = X ′ + h T, X i γ we get the equations ∇ T N = ∇ T ( N ′ + h T, N i γ ) = N ′′ + h T, ∇ T N i γ = N ′′ − k g γ, ∇ T T = k g N = γ ′′ + | γ ′ | γ = γ ′′ + γ. Combining the equations for ∇ T N we obtain N ′′ − k g γ = ( δ − δ ) N and rewriting this as an equation for γ yields the claim. (cid:3) Proposition 3.7.
Let γ : I → S be a curve on the three-dimensional sphere with the roundmetric. If k g = δ − δ the interpolating sesqui-harmonic curves are given by γ ( t ) = (cid:0) cos( p (1 − δ + δ ) t ) p (1 − δ + δ ) , sin( p (1 − δ + δ ) t ) p (1 − δ + δ ) , d , d (cid:1) (3.6) where − δ + δ + d + d = 1 .Proof. Making use of the assumptions (3.5) simplifies as γ ′′′′ + (1 − δ + δ ) γ ′′ = 0 . Solving this differential equation together with the constraints | γ | = 1 and | γ ′ | = 1 yields theclaim. (cid:3) Remark 3.8.
Note that it is required in (3.6) that 1 − δ + δ >
1, which is equivalent to δ > δ . This is consistent with the assumption k g = δ − δ . Theorem 3.9.
Let γ : I → S be a curve on the three-dimensional sphere with the round metricand suppose that δ > δ . Then the non-geodesic solution to (3.5) is given by γ ( t ) = 1 p a − a (cid:0)q − a cos( a t ) , q − a sin( a t ) , q a − a t ) , q a − a t ) (cid:1) (3.7) with the constants a := 1 √ r − δ + δ + q (1 + δ − δ ) + 4 k g ,a := 1 √ r − δ + δ − q (1 + δ − δ ) + 4 k g . Proof.
The most general ansatz for a solution of (3.5) is given by γ ( t ) = c cos( at ) + c sin( at ) + c cos( bt ) + c sin( bt ) , where c i , i = 1 . . . a and b are real numbers. This leads to thefollowing quadratic equation for both a and ba − a (1 − δ + δ ) + ( − k g − δ + δ ) = 0 . We obtain the two solutions a = 12 (cid:0) − δ + δ + q (1 + δ − δ ) + 4 k g (cid:1) ,a = 12 (cid:0) − δ + δ − q (1 + δ − δ ) + 4 k g (cid:1) . Moreover, the constraints | γ | = 1 and | γ ′ | = 1 give the two equations | c | + | c | = 1 , a | c | + a | c | = 1 . Solving this system for | c | and | c | yields the claim. (cid:3) N INTERPOLATING SESQUI-HARMONIC MAPS BETWEEN RIEMANNIAN MANIFOLDS 11
Remark 3.10.
If we analyze the constants appearing in (3.7) then we find that we have todemand the condition δ − δ > k g > a . In addition,we find a − a = q (1 + δ − δ ) + 4 k g > , − a = 12 (1 + δ − δ + q (1 + δ − δ ) + 4 k g ) > ,a − − − δ + δ + q (1 + δ − δ ) + 4 k g ) > δ , δ satisfying δ − δ > k g > Remark 3.11.
If we compare our results with [7, Theorem 3.3], then we find that interpolatingsesqui-harmonic curves on S have the same qualitative behavior as biharmonic curves. Moreprecisely, we have the following two cases(1) If k g = δ − δ , then γ is a circle of radius √ k g .(2) If δ − δ > k g >
0, then γ is a geodesic of the rescaled Clifford torus S (cid:0) r (1 + δ − δ + q (1 + δ − δ ) + 4 k g ) √ δ − δ ) + 4 k g ) (cid:1) × S (cid:0) r − − δ + δ + q (1 + δ − δ ) + 4 k g √ δ − δ ) + 4 k g ) (cid:1) . Note that the solutions from above reduce to biharmonic curves (see [7, Theorem 3.3]) in thecase of δ = 0 , δ = 1.We want to close this subsection by mentioning that it is possible to generalize the resultsobtained from above to higher-dimensional spheres as was done for biharmonic curves in [8].4. The qualitative behavior of solutions
In this section we study the qualitative behavior of interpolating sesqui-harmonic maps.In the case of a one-dimensional domain and the target being a Riemannian manifold, theEuler-Lagrange equation reduces to δ ∇ γ ′ γ ′ = δ R N ( γ ′ , ∇ γ ′ γ ′ ) γ ′ + δ ∇ γ ′ γ ′ , (4.1)where γ : I → N and γ ′ denotes the derivative with respect to the curve parameter s . Proposition 4.1.
Suppose that γ : I → N is a smooth solution of (1.4) . Then the followingconservation type law holds (cid:0) δ d ds − δ dds (cid:1) | γ ′ | = δ dds |∇ γ ′ γ ′ | . Proof.
We test (4.1) with γ ′ and obtain δ h∇ γ ′ γ ′ , γ ′ i = δ h∇ γ ′ γ ′ , γ ′ i = 12 δ dds | γ ′ | . The left-hand side can be further simplified as h∇ γ ′ γ ′ , γ ′ i = dds h∇ γ ′ γ ′ , γ ′ i − h∇ γ ′ γ ′ , ∇ γ ′ γ ′ i = d ds h∇ γ ′ γ ′ , γ ′ i − dds |∇ γ ′ γ ′ | = d ds h γ ′ , γ ′ i − dds |∇ γ ′ γ ′ | , which completes the proof. (cid:3) As already stated in the introduction it is obvious that harmonic maps solve (1.4). We will giveseveral conditions under which interpolating sesqui-harmonic maps must be harmonic general-izing several results from [19, 27]. To achieve these results we will frequently make use of thefollowing Bochner formula:
Lemma 4.2.
Let φ : M → N be a smooth solution of (1.4) . Then the following Bochner formulaholds ∆ 12 | τ ( φ ) | = |∇ τ ( φ ) | + h R N ( dφ ( e α ) , τ ( φ )) dφ ( e α ) , τ ( φ ) i + δ δ | τ ( φ ) | . (4.2) Proof.
This follows by a direct calculation. (cid:3)
Proposition 4.3.
Suppose that ( M, h ) is a compact Riemannian manifold. Let φ : M → N bea smooth solution of (1.4) . (1) If N has non-positive curvature K N ≤ and δ , δ have the same sign then φ is har-monic. (2) If | dφ | ≤ δ | R N | L ∞ δ and δ , δ have the same sign then φ is harmonic.Proof. The first statement follows directly from (4.2) by application of the maximum principle.For the second statement we estimate (4.2) as∆ 12 | τ ( φ ) | = |∇ τ ( φ ) | + ( δ δ − | R N | L ∞ | dφ | ) | τ ( φ ) | ≥ (cid:3) If we do not require M to be compact we can give the following result. Proposition 4.4.
Let φ : M → N be a Riemannian immersion that solves (1.4) with | τ ( φ ) | = const . If N has non-positive curvature K N ≤ and δ , δ have the same sign then φ must beharmonic.Proof. Via the maximum principle we obtain ∇ τ ( φ ) = 0 from (4.2). By assumption the map φ is an immersion such that −| τ ( φ ) | = h dφ, ∇ τ ( φ ) i concluding the proof. (cid:3) In the case that dim M = dim N − N having negative sectional curvaturecan be replaced by demanding negative Ricci curvature. Theorem 4.5.
Let φ : M → N be a Riemannian immersion. Suppose that M is compact and dim M = dim N − . If N has non-positive Ricci curvature and δ , δ have the same sign then φ is interpolating sesqui-harmonic if and only if it is harmonic.Proof. Since φ is an immersion and dim M = dim N − R N ( dφ ( e α ) , τ ( φ )) dφ ( e α ) = − Ric N ( τ ( φ )) . Making use of (4.2) we get∆ 12 | τ ( φ ) | = |∇ τ ( φ ) | − h Ric N ( τ ( φ )) , τ ( φ ) i + δ δ | τ ( φ ) | ≥ (cid:3) As for harmonic maps ([29, Theorem 2]) we can prove a unique continuation theorem for inter-polating sesqui-harmonic maps. To obtain this result we recall the following ([1, p.248])
Theorem 4.6.
Let A be a linear elliptic second-order differential operator defined on a domain D of R n . Let u = ( u , . . . , u n ) be functions in D satisfying the inequality | Au j | ≤ C (cid:0) X α,i (cid:12)(cid:12) ∂u i ∂x α (cid:12)(cid:12) + X i | u i | (cid:1) . (4.3) If u = 0 in an open set, then u = 0 throughout D . N INTERPOLATING SESQUI-HARMONIC MAPS BETWEEN RIEMANNIAN MANIFOLDS 13
Making use of this result we can prove the following
Proposition 4.7.
Let φ ∈ C ( M, N ) be an interpolating sesqui-harmonic map. If φ is harmonicon a connected open set W of M then it is harmonic on the whole connected component of M which contains W .Proof. The analytic structure of the interpolating sesqui-harmonic map equation is the following | ∆ τ ( φ ) | ≤ C ( | dφ | | τ ( φ ) | + | τ ( φ ) | ) . In order to apply Theorem 4.6 we consider the equation for interpolating sesqui-harmonic mapsin a coordinate chart in the target. The bound on | dφ | can be obtained by shrinking the chartif necessary such that (4.3) holds. (cid:3) Interpolating sesqui-harmonic maps with vanishing energy-momentum tensor.
In this section we study the qualitative behavior of solutions to (1.4) under the additionalassumption that the energy-momentum tensor (2.2) vanishes similar to [22]. Such an assumptionis partially motivated from physics: In physics one usually also varies the action functional (1.3)with respect to the metric on the domain and the resulting Euler-Lagrange equation yields thevanishing of the energy-momentum tensor.In the following we will often make use of the trace of the energy-momentum tensor (2.2), whichis given by (where n = dim M )Tr T = δ (1 − n | dφ | + δ n | τ ( φ ) | + δ ( n − h dφ, ∇ τ ( φ ) i . (4.4)Note that we do not have to assume that φ is a solution of (1.4) in the following. Proposition 4.8.
Let γ : S → N be a curve with vanishing energy-momentum tensor. If δ δ > then γ maps to a point.Proof. Using (4.4) and integrating over S we find0 = δ Z S | γ ′ | ds + 32 δ Z S | τ ( γ ) | ds, which yields the claim. (cid:3) Proposition 4.9.
Suppose that ( M, h ) is a Riemannian surface. Let φ : M → N be a smoothmap with vanishing energy-momentum tensor. Then φ is harmonic.Proof. Since dim M = 2 we obtain from (4.4) that | τ ( φ ) | = 0 yielding the claim. (cid:3) For a higher-dimensional domain we have the following result.
Proposition 4.10.
Let φ : M → N be a smooth map with vanishing energy-momentum tensor.Then the following statements hold: (1) If dim M = 3 and δ δ < then φ is trivial. (2) If dim M = 4 then φ is trivial. (3) If dim M ≥ and δ δ > then φ is trivial.Proof. Integrating (4.4) we obtain0 = δ (1 − n Z M | dφ | dV + δ (2 − n Z | τ ( φ ) | dV, which already yields the result. (cid:3) As a next step we rewrite the condition on the vanishing of the energy-momentum tensor.
Proposition 4.11.
Let φ : M → N be a smooth map and assume that n = 2 . Then thevanishing of the energy-momentum tensor is equivalent to T ( X, Y ) = δ h dφ ( X ) , dφ ( Y ) i (4.5) − δ n − | τ ( φ ) | h ( X, Y ) − δ (cid:0) h dφ ( X ) , ∇ Y τ ( φ ) i + h dφ ( Y ) , ∇ X τ ( φ ) i (cid:1) . Proof.
Rewriting the equation for the vanishing of the trace of the energy-momentum tensor(4.4) we find δ h dφ, ∇ τ ( φ ) i = δ ( n − n − | dφ | − δ n n − | τ ( φ ) | . Inserting this into the energy-momentum-tensor (2.2) yields the claim. (cid:3)
This allows us to give the following
Proposition 4.12.
Let φ : M → N be a smooth map with vanishing energy-momentum tensor.Suppose that dim M > and rank φ ≤ n − . Then φ is harmonic.Proof. Fix a point p ∈ M . By assumption rank φ ≤ n − X p ∈ ker dφ p . For X = Y = X p we can infer from (4.5) that τ ( φ ) = 0 yielding the claim. (cid:3) If the domain manifold M is non-compact we can give the following variant of the previousresults. Proposition 4.13.
Let φ : M → N be a smooth Riemannian immersion with vanishing energy-momentum tensor. If dim M = 2 then φ is harmonic, if dim M = 4 then φ is trivial.Proof. Since φ is a Riemannian immersion, we have h τ ( φ ) , dφ i = 0. Hence (4.4) yields0 = δ (1 − n | dφ | + δ (2 − n | τ ( φ ) | , which proves the claim. (cid:3) Conformal construction of interpolating sesqui-harmonic maps.
In [3] the authorspresent a powerful construction method for biharmonic maps. Instead of trying to directly solvethe fourth-order equation for biharmonic maps they assume the existence of a harmonic mapand then perform a conformal transformation of the metric on the domain to render this mapbiharmonic. In particular, they call a metric that renders the identity map biharmonic, a biharmonic metric . In this section we will discuss if the same approach can also be used toconstruct interpolating sesqui-harmonic maps.To this end let φ : ( M, h ) → ( N, g ) be a smooth map. If we perform a conformal transformationof the metric on the domain, that is ˜ h = e u h for some smooth function u , we have the followingformula for the transformation of the tension field τ ˜ h ( φ ) = e − u ( τ h ( φ ) + ( n − dφ ( ∇ u )) , where τ ˜ h ( φ ) denotes the tension field of the map φ with respect to the metric ˜ h . In addition,we set n := dim M .Now, suppose that φ is a harmonic map with respect to h , that is τ h ( φ ) = 0, then we obtainthe identity τ ˜ h ( φ ) = e − u ( n − dφ ( ∇ u ) . This allows us to deduce
Proposition 4.14.
Let φ : ( M, h ) → ( N, g ) be a smooth harmonic map and suppose that dim M = 2 . Let ˜ h = e u h be a metric conformal to h . Then the map φ : ( M, ˜ h ) → ( N, g ) is interpolating sesqui-harmonic if and only if δ (cid:0) ∇ ∗ ∇ dφ ( ∇ u ) + ( n − ∇ ∇ u dφ ( ∇ u ) + 2( − ∆ u − ( n − | du | ) dφ ( ∇ u )+ Tr h R N ( dφ ( ∇ u ) , dφ ) dφ (cid:1) = δ e − u dφ ( ∇ u ) . Proof.
For every v ∈ Γ( φ ∗ T N ) the following formula holds ∇ ∗ ˜ h ∇ ˜ h v = e − u ( ∇ ∗ h ∇ h v + ( n − ∇ ∇ u v ) . By a direct calculation we find ∇ ∗ h ∇ h ( τ ˜ h ( φ )) =( n − e − u (cid:0) − udφ ( ∇ u ) + 4 | du | dφ ( ∇ u ) − h∇ u, ∇ ( dφ ( ∇ u )) i + ∇ ∗ ∇ dφ ( ∇ u ) (cid:1) . N INTERPOLATING SESQUI-HARMONIC MAPS BETWEEN RIEMANNIAN MANIFOLDS 15
Together with ∇ ∇ u τ ˜ h ( φ ) = ( n − e − u ( − |∇ u | dφ ( ∇ u ) + ∇ ∇ u dφ ( ∇ u )) ,R N ( dφ, τ ˜ h ( φ )) dφ = ( n − e − u R N ( dφ, dφ ( ∇ u )) dφ this completes the proof. (cid:3) In the following we will call a metric that renders the identity map interpolating sesqui-harmonican interpolating sesqui-harmonic metric . Corollary 4.15.
Let φ : ( M, h ) → ( M, h ) be the identity map and suppose that dim M = 2 .Let ˜ h = e u h be a metric conformal to h . Then the map φ : ( M, ˜ h ) → ( M, h ) is interpolatingsesqui-harmonic if and only if δ (2( − ∆ u − ( n − | du | ) ∇ u + ( n − ∇ ∇ u ∇ u + Tr h ( ∇ ∗ ∇ ) ∇ u + Ric M ( ∇ u )) = δ e − u ∇ u. We now rewrite this as an equation for ∇ u . Proposition 4.16.
Let φ : ( M, h ) → ( M, h ) be the identity map and suppose that dim M = 2 .Let ˜ h = e u h be a metric conformal to h and set ∇ u = β . Then the map φ : ( M, ˜ h ) → ( M, h ) isinterpolating sesqui-harmonic if and only if − ∆ β = 2( d ∗ β − ( n − | β | ) β + ( n −
6) 12 d | β | + 2 Ric M ( β ♯ ) ♭ − δ δ e − u β, (4.6) where − ∆ = dd ∗ + d ∗ d is the Laplacian acting on one-forms.Proof. The Laplacian acting on one-forms satisfies the following Weitzenb¨ock identity∆ β ( X ) = Tr h ( ∇ ) β ( X ) − β Ric( X ) , where X is a vector field. In addition, note that (see [3, Proof of Proposition 2.2] for moredetails) ∇ ∇ u ∇ u = 12 d | β | , which completes the proof. (cid:3) Proposition 4.17.
Let ( M, h ) be a compact manifold of strictly negative Ricci curvature with dim M > and assume that δ δ > . Then there does not exist an interpolating sesqui-harmonic metric that is conformally related to h except a constant multiple of h .Proof. Note that our sign convention for the Laplacian is different from the one used in [3]. Wedefine the one-form θ := e − u β . By a direct calculation we then find using (4.6)∆ θ = −
12 ( n − e u d | θ | − M ( θ ♯ ) ♭ + δ δ e − u θ. In addition, we have∆ 12 | θ | = h ∆ θ, θ i + |∇ θ | + Ric( θ ♯ , θ ♯ )= −
12 ( n − e u h d | θ | , θ i + |∇ θ | − Ric( θ ♯ , θ ♯ ) + δ δ e − u | θ | . The claim then follows by the maximum principle. (cid:3)
Remark 4.18.
In contrast to the case of biharmonic maps (4.6) contains also a term involving u on the right hand side. This reflects the fact that both harmonic and biharmonic maps on its ownhave a nice behavior under conformal deformations of the domain metric, whereas interpolatingsesqui-harmonic maps do not. This prevents us from making a connection between interpolatingsesqui-harmonic metrics and isoparametric functions as was done in [3] for biharmonic maps. A Liouville-type theorem for interpolating sesqui-harmonic maps between com-plete manifolds.
In this section we will prove a Liouville-type theorem for solutions of (1.4)between complete Riemannian manifolds generalizing a similar result for biharmonic maps from[25]. For more Liouville-type theorems for biharmonic maps see [5, 6] and references therein.To this end we will make use of the following result due to Gaffney [14]:
Theorem 4.19.
Let ( M, h ) be a complete Riemannian manifold. If a C one-form ω satisfies Z M | ω | dV < ∞ and Z M | δω | dV < ∞ or, equivalently, a C vector field X defined by ω ( Y ) = h ( X, Y ) , satisfies Z M | X | dV < ∞ and Z M div( X ) dV < ∞ , then Z M ( δω ) dV = Z M div( X ) dV = 0 . Theorem 4.20.
Let ( M, h ) be a complete non-compact Riemannian manifold and ( N, g ) amanifold with non-positive sectional curvature. Let φ : M → N be a smooth solution of (1.4) and p be a real constant satisfying ≤ p < ∞ . (1) If δ δ > and Z M | τ ( φ ) | p dV < ∞ , Z M | dφ | dV < ∞ then φ must be harmonic. (2) If δ δ > , vol( M, h ) = ∞ and Z M | τ ( φ ) | p dV < ∞ then φ must be harmonic.Proof. We choose a cutoff function 0 ≤ η ≤ M that satisfies η ( x ) = 1 for x ∈ B R ( x ) , η ( x ) = 0 for x ∈ B R ( x ) , |∇ η | ≤ CR for x ∈ M, where B R ( x ) denotes the geodesic ball around the point x with radius R .We test the interpolating sesqui-harmonic map equation (1.4) with η τ ( φ ) | τ ( φ ) | p − and find η | τ ( φ ) | p − h ∆ τ ( φ ) , τ ( φ ) i = η | τ ( φ ) | p − h R N ( dφ ( e α ) , τ ( φ )) dφ ( e α ) , τ ( φ ) i + δ δ η | τ ( φ ) | p ≥ , where we made use of the assumptions on the curvature of the target and the signs of δ and δ . Integrating over M and using integration by parts we obtain Z M η | τ ( φ ) | p − h ∆ τ ( φ ) , τ ( φ ) i dV = − Z M h∇ τ ( φ ) , τ ( φ ) i| τ ( φ ) | p − η ∇ η dV − ( p − Z M η |h∇ τ ( φ ) , τ ( φ ) i| | τ ( φ ) | p − dV − Z M η |∇ τ ( φ ) | | τ ( φ ) | p − dV ≤ CR Z M | τ ( φ ) | p dV − Z M η |∇ τ ( φ ) | | τ ( φ ) | p − dV − ( p − Z M η |h∇ τ ( φ ) , τ ( φ ) i| | τ ( φ ) | p − dV, N INTERPOLATING SESQUI-HARMONIC MAPS BETWEEN RIEMANNIAN MANIFOLDS 17 where we used Young’s inequality and the properties of the cutoff function η . Combining bothequations we find12 Z M η |∇ τ ( φ ) | | τ ( φ ) | p − dV ≤ CR Z M | τ ( φ ) | p dV − ( p − Z M η |h∇ τ ( φ ) , τ ( φ ) i| | τ ( φ ) | p dV. Letting R → ∞ and using the finiteness assumption of the L p norm of the tension field we maydeduce that τ ( φ ) is parallel and thus has constant norm.To establish the first claim of the theorem we make use of Theorem 4.19. We define a one-form ω by ω ( X ) := | τ ( φ ) | p − h dφ ( X ) , τ ( φ ) i , where X is a vector field on M . Note that Z M | ω | dV ≤ Z M | dφ || τ ( φ ) | p dV ≤ (cid:0) Z M | dφ | dV (cid:1) (cid:0) Z M | τ ( φ ) | p dV (cid:1) < ∞ . Using that | τ ( φ ) | has constant norm we find by a direct calculation that δω = | τ ( φ ) | p +1 .Again, since | τ ( φ ) | has constant norm and the L p -norm of τ ( φ ) is bounded, we find that | δω | isintegrable over M . By application of Theorem 4.19 we can then deduce that τ ( φ ) = 0.To prove the second claim, we note that vol( M, h ) = ∞ and | τ ( φ ) | 6 = 0 give Z M | τ ( φ ) | p dV = | τ ( φ ) | p vol( M, h ) = ∞ , yielding a contradiction. (cid:3) Acknowledgements:
The author gratefully acknowledges the support of the Austrian ScienceFund (FWF) through the project P30749-N35 “Geometric variational problems from stringtheory”.
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University of Vienna, Faculty of Mathematics, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria,
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