Nonlinear Dirac equations, Monotonicity Formulas and Liouville Theorems
aa r X i v : . [ m a t h . DG ] M a y NONLINEAR DIRAC EQUATIONS, MONOTONICITY FORMULAS ANDLIOUVILLE THEOREMS
VOLKER BRANDING
Abstract.
We study the qualitative behavior of nonlinear Dirac equations arising in quantumfield theory on complete Riemannian manifolds. In particular, we derive monotonicity formulasand Liouville theorems for solutions of these equations. Finally, we extend our analysis to Dirac-harmonic maps with curvature term. Introduction and results
In quantum field theory spinors are employed to model fermions. The equations that governthe behavior of fermions are both linear and nonlinear Dirac equations. A Dirac equation withvanishing right hand side describes a free massless fermion and linear Dirac equations describefree fermions having a mass. However, to model the interaction of fermions one has to take intoaccount nonlinearities.In mathematical terms spinors are sections in a vector bundle, the spinor bundle, which isdefined on a Riemannian spin manifold. The spin condition is of topological nature and ensuresthe existence of the spinor bundle Σ M . The mathematical analysis of linear and nonlinearDirac equations comes with two kinds of difficulties: First of all, the Dirac operator is of firstorder, such that tools like the maximum principle are not available. Secondly, in contrast tothe Laplacian, the Dirac operator has its spectrum on the whole real line.Below we give a list of energy functionals that arise in quantum field theory. Their criticalpoints all lead to nonlinear Dirac equations. To this end let D be the classical Dirac operatoron a Riemannian spin manifold ( M, g ) of dimension n and e i an orthonormal basis of T M .Furthermore, let · be the Clifford multiplication of spinors with tangent vectors and ω C thecomplex volume form. Moreover, we fix a hermitian scalar product on the spinor bundle.(1) The Soler model [27] describes fermions that interact by a quartic term in the actionfunctional. In quantum field theory this model is usually studied on four-dimensionalMinkowski space: E ( ψ ) = Z M ( h ψ, Dψ i − λ | ψ | − µ | ψ | ) d vol g (2) The Thirring model [28] describes the self-interaction of fermions in two-dimensionalMinkowski space: E ( ψ ) = Z M ( h ψ, Dψ i − λ | ψ | − µ n X i =1 h ψ, e i · ψ ih ψ, e i · ψ i ) d vol g (3) The Nambu–Jona-Lasinio model [25] is a model for interacting fermions with chiral sym-metry. It also contains a quartic interaction term and is defined on an even-dimensionalspacetime: E ( ψ ) = Z M (cid:0) h ψ, Dψ i + µ | ψ | − h ψ, ω C · ψ ih ψ, ω C · ψ i ) (cid:1) d vol g Note that this model does not have a term proportional to | ψ | in the energy functional. Date : October 24, 2018.2010
Mathematics Subject Classification.
Key words and phrases. nonlinear Dirac equations; monotonicity formulas; Liouville Theorems; Dirac-harmonic maps with curvature term. (4) The
Gross–Neveu model with N flavors [17] is a model for N interacting fermions intwo-dimensional Minkowski space: E ( ψ ) = Z M ( h ψ, Dψ i − λ | ψ | + µ N | ψ | ) d vol g The spinors that we are considering here are twisted spinors, more precisely ψ ∈ Γ(Σ M ⊗ R N ).(5) The nonlinear supersymmetric sigma model consists of a map φ between two Riemannianmanifolds M and N and a vector spinor ψ ∈ Γ(Σ M ⊗ φ ∗ T N ). Moreover, R N is thecurvature tensor on N and /D denotes the Dirac operator acting on vector spinors. Theenergy functional under consideration is E c ( φ, ψ ) = Z M ( | dφ | + h ψ, /Dψ i − h R N ( ψ, ψ ) ψ, ψ i ) d vol g . The critical points of this functional became known in the mathematical literature as
Dirac-harmonic maps with curvature term .In the models (1)-(4) from above the real parameter λ can be interpreted as mass, whereas thereal constant µ describes the strength of interaction. All of the models listed above lead tononlinear Dirac equations of the form Dψ ∼ λψ + µ | ψ | ψ. (1.1)Note that in the physical literature Clifford multiplication is usually expressed as matrix mul-tiplication with γ µ and the complex volume element is referred to as γ . In contrast to thephysical literature we will always assume that spinors are commuting, whereas in the physicalliterature they are mostly assumed to be Grassmann-valued. For simplicity we will mainly focuson the Soler model.Several existence results for equations of the form (1.1) are available: In [18] existence resultsfor nonlinear Dirac equations on compact spin manifolds are obtained. For n ≥ Dψ = λψ + | ψ | n − ψ with λ ∈ R , have been obtained in [19]. For λ = 0 this equation is known as the spinorialYamabe equation. In particular, this equation is interesting for n = 2 since it is closely relatedto conformally immersed constant mean curvature surfaces in R . Moreover, existence resultsfor the spinorial Yamabe equation have been obtained on S [21] and on S n [20] for n ≥ Nonlinear Dirac equations on Riemannian manifolds
Let (
M, g ) be a Riemannian spin manifold of dimension n . A Riemannian manifold admitsa spin structure if the second Stiefel-Whitney class of its tangent bundle vanishes.We briefly recall the basic notions from spin geometry, for a detailed introduction to spingeometry we refer to the book [23].We fix a spin structure on the manifold M and consider the spinor bundle Σ M . On thespinor bundle Σ M we have the Clifford multiplication of spinors with tangent vectors denoted ONLINEAR DIRAC EQUATIONS, MONOTONICITY FORMULAS AND LIOUVILLE THEOREMS 3 by · . Moreover, we fix a hermitian scalar product on the spinor bundle and denote its real partby h· , ·i . Clifford multiplication is skew-symmetric h ψ, X · ξ i = −h X · ψ, ξ i for all ψ, ξ ∈ Γ(Σ M ) and X ∈ T M . Moreover the Clifford relations X · Y + Y · X = − g ( X, Y ) (2.1)hold for all
X, Y ∈ T M . The Dirac operator D : Γ(Σ M ) → Γ(Σ M ) is defined as the compositionof first applying the covariant derivative on the spinor bundle followed by Clifford multiplication.More precisely, it is given by D := n X i =1 e i · ∇ Σ Me i , where e i , i = 1 . . . n is an orthonormal basis of T M . Sometimes we will make use of the Einsteinsummation convention and just sum over repeated indices. The Dirac operator is of first order,elliptic and self-adjoint with respect to the L -norm. Hence, if M is compact the Dirac operatorhas a real and discrete spectrum.The square of the Dirac operator satisfies the Schroedinger-Lichnerowicz formula D = ∇ ∗ ∇ + R , (2.2)where R denotes the scalar curvature of the manifold M .After having recalled the basic definition from spin geometry will focus on the analysis of thefollowing action functional (which is the first one from the introduction) E ( ψ ) = Z M ( h ψ, Dψ i − λ | ψ | − µ | ψ | ) d vol g . (2.3)Its critical points are given by Dψ = λψ + µ | ψ | ψ. (2.4)It turns out that L (Σ M ) × W , (Σ M ) is the right function space for weak solutions of (2.4). Definition 2.1.
We call ψ ∈ L (Σ M ) × W , (Σ M ) a weak solution if it solves (2.4) in adistributional sense.The analytic structure of the other energy functionals listed in the introduction is the sameas the one of (2.3). Due to this reason many of the results that will be obtained for solutionsof (2.4) can easily be generalized to critical points of the other models.The equation (2.4) is also interesting from a geometric point of view since it interpolatesbetween eigenspinors ( µ = 0) and a non-linear Dirac equation ( λ = 0) that arises in the studyof CMC immersion from surfaces into R .In the following we want to vary the energy functional (2.3) with respect to the metric g .To this end let us recall the following construction for identifying spinor bundles belonging todifferent metrics. For the Riemannian case this was established in [5] and later on generalizedto the pseudo-Riemannian case in [4]. Here, we will follow the presentation from [22], Chapter2. Suppose we have two spinor bundles Σ g M and Σ h M corresponding to different metrics g and h . There exists a unique positive definite tensor field h g uniquely determined by therequirement h ( X, Y ) = g ( HX, HY ) = g ( X, h g Y ), where H := p h g . Let P g and P h be theoriented orthonormal frame bundles of ( M, g ) and (
M, h ). Then H − induces an equivariantisomorphism b g,h : P g → P h via the assignment E i H − E i , i = 1 . . . n . We fix a spin structureΛ g : Q g → P g of ( M, g ) and think of it as a Z -bundle. The pull-back of Λ g via the isomorphism b h,g : P h → P g induces a Z -bundle Λ h : Q h → P h . Moreover, we get a Spin( n )-equivariantisomorphism ˜ b h,g : Q h → Q g such that the following diagram commutes: VOLKER BRANDING Q h Λ h (cid:15) (cid:15) ˜ b h,g / / Q g Λ g (cid:15) (cid:15) P h b h,g / / P g Making use of this construction we obtain the following
Lemma 2.2.
There exist natural isomorphisms b g,h : T M → T M, β g,h : Σ g M → Σ h M that satisfy h ( b h,g X, b h,g Y ) = g ( X, Y ) , h β h,g χ, β h,g ψ i Σ h M = h ψ, χ i Σ g M , ( b g,h X ) · ( β g,h ψ ) = ˜ β g,h ( X · ψ ) for all X, Y ∈ Γ( T M ) and ψ, χ ∈ Γ(Σ g M ) . In order to calculate the variation of the Dirac operator with respect to the metric we needthe following objects: Let Sym(0 ,
2) be the space of all symmetric (0 , M, g ).Any element k of Sym(0 ,
2) induces a (1 , k g via k ( X, Y ) = g ( X, k g Y ). We denotethe Dirac operator on ( M, g + tk ) by D g + tk for a small parameter t . Moreover, we will use thenotation ψ g + tk := β g,g + tk ψ ∈ Γ(Σ M g + tk ), which can be thought of as push-forward of ψ ∈ Σ g M to ψ ∈ Σ g + tk M . Lemma 2.3.
The variation of the Dirac-energy with respect to the metric is given by ddt (cid:12)(cid:12) t =0 h ψ g + tk , D g + tk ψ g + tk i Σ g + tk M = − h e i · ∇ Σ Me j ψ + e j · ∇ Σ Me i ψ, ψ i Σ g M k ij , (2.5) where the tensor on the right hand side is the stress-energy tensor associated to the Dirac energy. Definition 2.4.
A weak solution ψ ∈ L (Σ M ) × W , (Σ M ) is called stationary if it is also acritical point of E ( ψ ) with respect to domain variations. Proposition 2.5.
A stationary solution ψ ∈ L (Σ M ) × W , (Σ M ) of (2.4) satisfies Z M ( h e i · ∇ Σ Me j ψ + e j · ∇ Σ Me i ψ, ψ i − g ij µ | ψ | ) k ij d vol g , (2.6) where k ij is a smooth element of Sym(0 , .Proof. We calculate ddt (cid:12)(cid:12) t =0 Z M (cid:0) h ψ g + tk , D g + tk ψ g + tk i Σ g + tk M − λ | ψ g + tk | g + tk M − µ | ψ g + tk | g + tk M (cid:1) d vol g + tk , where k is a symmetric (0 , t some small number. The variation of the volume-element yields ddt (cid:12)(cid:12) t =0 d vol g + tk = 12 h g, k i g d vol g . (2.7)Using that β g,g + tk acts as an isometry on the spinor bundle we obtain | ψ g + tk | g + tk M = | β g,g + tk ψ | g + tk M = | ψ | g M and together with (2.5) the result follows. (cid:3) For a smooth solution ψ of (2.4) we thus obtain the stress-energy tensor S ij = h e i · ∇ Σ Me j ψ + e j · ∇ Σ Me i ψ, ψ i − g ij µ | ψ | . (2.8)Its trace can easily be computed to betr S = g ij S ij = 2 λ | ψ | + (2 − n ) µ | ψ | . Note that the stress-energy tensor is traceless for λ = 0 and n = 2 since it arises from aconformally invariant energy functional in that case. ONLINEAR DIRAC EQUATIONS, MONOTONICITY FORMULAS AND LIOUVILLE THEOREMS 5
Lemma 2.6.
Suppose that ψ is a smooth solution of (2.4) . Then the stress-energy tensor (2.8) is symmetric and divergence-free.Proof. We choose a local orthonormal basis of
T M such that ∇ e i e j = 0 , i, j = 1 , . . . , n at theconsidered point. To show that the stress-energy tensor is divergence-free we calculate ∇ j S ij = ∇ j ( h e i · ∇ e j ψ + e j · ∇ e i ψ, ψ i − g ij µ | ψ | )= h e i · ∆ ψ, ψ i + h e i · ∇ e j ψ, ∇ e j ψ i | {z } =0 + h D ∇ e i ψ, ψ i − h∇ e i ψ, Dψ i − µ | ψ | h∇ e i ψ, ψ i . By a direct computation we find h D ∇ e i ψ, ψ i = h ψ, e j · R Σ M ( e i , e j ) ψ i | {z } = h ψ, Ric( e i ) · ψ i =0 + h∇ e i Dψ, ψ i = ( λ + 3 µ | ψ | ) h∇ e i ψ, ψ i , h∇ e i ψ, Dψ i =( λ + µ | ψ | ) h∇ e i ψ, ψ i , where we used that ψ is a solution of (2.4). Thus, we obtain ∇ j S ij = h e i · ∆ ψ, ψ i − µ | ψ | h∇ e i ψ, ψ i . Using (2.2) and (2.4) we find that h e i · ∆ ψ, ψ i = − µ h e i · ( ∇| ψ | ) · ψ, ψ i = µg ( e i , ∇| ψ | ) | ψ | = 2 µ | ψ | h∇ e i ψ, ψ i , which completes the proof. (cid:3) We will often make use of the following Bochner-type equation
Lemma 2.7.
Let ψ be a smooth solution of (2.4) . Then the following formula holds ∆ 12 | ψ | = (cid:12)(cid:12) d | ψ | (cid:12)(cid:12) + | ψ | |∇ ψ | + | ψ | (cid:0) R − ( λ + µ | ψ | ) (cid:1) . (2.9) Proof.
By a direct calculation we find∆ 12 | ψ | = (cid:12)(cid:12) d | ψ | (cid:12)(cid:12) + | ψ | |∇ ψ | + R | ψ | − | ψ | h ψ, D ψ i , where we used (2.2). Moreover, we obtain h ψ, D ψ i = h ψ, D ( λψ ) i + h ψ, D ( µ | ψ | ψ ) i = λ | ψ | + λµ | ψ | + µ h ψ, ( ∇| ψ | ) · ψ i | {z } =0 + µλ | ψ | + µ | ψ | = | ψ | ( λ + µ | ψ | ) , (2.10)where we used that ψ is a solution of (2.4). (cid:3) Let us recall the following definitions:
Definition 2.8.
A spinor ψ ∈ Γ(Σ M ) is called twistor spinor if it satisfies ∇ Σ MX ψ + 1 n X · Dψ = 0 (2.11)for all vector fields X . The spinor ψ is called Killing spinor if it is both a twistor spinor andan eigenspinor of the Dirac operator, that is ∇ Σ MX ψ + αX · ψ = 0 (2.12)with α ∈ R .It is well known that Killing spinors have constant norm, that is | ψ | = const . However, herewe have the following Lemma 2.9.
Suppose that ψ is a solution of (2.4) and a twistor spinor. Then ψ has constantnorm. VOLKER BRANDING
Proof.
We calculate for an arbitrary X ∈ T M∂ X | ψ | = h∇ Σ MX ψ, ψ i = − n h X · Dψ, ψ i = − n ( λ + µ | ψ | ) h X · ψ, ψ i , where we first used that ψ is a twistor spinor and then used that ψ is a solution of (2.4). Thestatement then follows from the skew-symmetry of the Clifford multiplication. (cid:3) Example 2.10.
Suppose that ψ is a Killing spinor with constant α = λ + µ | ψ | n . Then it is asolution of (2.4). However, this above approach is rather restrictive since only few Riemannianmanifolds admit Killing spinors [2]. Proposition 2.11.
Suppose that ψ is a smooth solution of (2.4) and also a twistor spinor.Then the stress-energy tensor (2.8) acquires the form S ij = 1 n g ij (cid:0) µ (2 − n ) | ψ | + 2 λ | ψ | (cid:1) . (2.13)In particular, the stress-energy tensor is just a multiple of the metric. Proof.
We consider the stress-energy tensor (2.8) and use the fact that ψ is a twistor spinor,that is S ij = h e i · ∇ e j ψ + e j · ∇ e i ψ, ψ i − g ij µ | ψ | = − n h ( e i · e j + e j · e i | {z } = − g ij ) Dψ, ψ i − g ij µ | ψ | = µ ( 2 n − | ψ | g ij + 2 n λ | ψ | g ij , where we used the Clifford relations (2.1) and (2.4). (cid:3) Nonlinear Dirac equations on closed manifolds
In this section we will derive several properties of solutions of (2.4) on closed Riemannianmanifolds, where we will mostly focus on the two-dimensional case.3.1.
Nonlinear Dirac equations on closed surfaces.
First, we derive a local energy es-timates for smooth solutions of (2.4). Our result is similar to the energy estimate that wasobtained in [14], Theorem 2.1, which corresponds to (2.4) with λ = 0. We obtain the following Theorem 3.1.
Let ψ be a smooth solution of (2.4) . If | ψ | L ( D ) < ǫ then | ψ | W k,p ( D ′ ) ≤ C | ψ | L ( D ) (3.1) for all D ′ ⊂ D and p > . The constant C depends on D ′ , µ, λ, k, p . We will divide the proof into two Lemmata, the result then follows by iterating the procedureoutlined below.
Lemma 3.2.
Let ψ be a smooth solution of (2.4) . If | ψ | L ( D ) < ǫ then for all p > and all D ′ ⊂ D we have | ψ | L p ( D ′ ) ≤ C | ψ | L ( D ) , (3.2) where the constant C depends on D ′ , µ, λ, k, p .Proof. Choose a cut-off function η with 0 ≤ η ≤ η | D ′ = 1 and supp η ⊂ D . Then we have D ( ηψ ) = ηDψ + ∇ η · ψ = ηλψ + ηµ | ψ | ψ + ∇ η · ψ. We set ξ = ηψ and by making use of elliptic estimates for first order equations we obtain | ξ | W ,q ( D ) ≤ C ( | ηψ | L q ( D ) + µ (cid:12)(cid:12) | ψ η | (cid:12)(cid:12) L q ( D ) + |∇ η || ψ | L q ( D ) ) ≤ C ( | ψ | L q ( D ) + (cid:12)(cid:12) | ψ η | (cid:12)(cid:12) L q ( D ) ) . We set q ∗ := q − q for q <
2. By the H¨older inequality we get (cid:12)(cid:12) | ψ η | (cid:12)(cid:12) L q ( D ) ≤ | ψ | L ( D ) | ξ | L q ∗ ( D ) . ONLINEAR DIRAC EQUATIONS, MONOTONICITY FORMULAS AND LIOUVILLE THEOREMS 7
Applying the Sobolev embedding theorem in two dimensions we find | ξ | L q ∗ ( D ) ≤ C | ξ | W ,q ( D ) ≤ C ( | ψ | L q ( D ) + | ψ | L ( D ) | ξ | L q ∗ ( D ) ) . Using the small energy assumption we get | ξ | L q ∗ ( D ) ≤ C | ψ | L ( D ) . For any p > q < p = q ∗ . (cid:3) Lemma 3.3.
Let ψ be a smooth solution of (2.4) . If | ψ | L ( D ) < ǫ then for all p > and all D ′ ⊂ D we have | ψ | W ,p ( D ′ ) ≤ C | ψ | L ( D ) , (3.3) where the constant C depends on D ′ , µ, λ, k, p .Proof. Again, choose a cut-off function η with 0 ≤ η ≤ η | D ′ = 1 and supp η ⊂ D . Setting ξ = ηψ we locally have Z D |∇ ξ | dx = Z D | Dξ | dx = Z D | ηλψ + ηµ | ψ | ψ + ∇ η · ψ | dx ≤ C Z D ( | ψ | + | ψ | ) dx. We obtain the following inequality |∇ ξ | L ( D ) ≤ C ( | ψ | L ( D ) + | ψ | L ) ≤ C | ψ | L ( D ) , which yields |∇ ψ | L ( D ′ ) ≤ C | ψ | L ( D ) . (3.4)By a direct computation we find D ψ = λ ψ + 2 µλ | ψ | ψ + µ | ψ | ψ + µ ( ∇| ψ | ) · ψ and also ∆ ξ = (∆ η ) ψ + 2 ∇ η ∇ ψ + η ∆ ψ. This yields | ∆ ξ | ≤ C ( | ψ | + |∇ ψ | + | ψ | |∇ ψ | + | ψ | + | ψ | ) . On the disc D we have ∆ = − D , hence we find | ηψ | W ,p ( D ) ≤ C ( | ψ | L p ( D ) + |∇ ψ | L p ( D ) + (cid:12)(cid:12) | ψ | |∇ ψ | (cid:12)(cid:12) L p ( D ) + (cid:12)(cid:12) | ψ | (cid:12)(cid:12) L p ( D ) + (cid:12)(cid:12) | ψ | (cid:12)(cid:12) L p ( D ) ) . (3.5)Using (3.2) and (3.4) we obtain (cid:12)(cid:12) | ψ | |∇ ψ | (cid:12)(cid:12) L p ( D ) ≤ C |∇ ψ | L ( D ′ ) | ψ | L ( D ′ ) ≤ C | ψ | L ( D ) and the same bound applies to the first and the last two terms of (3.5). Thus, we obtain bysetting p = in (3.2) and applying the Sobolev embedding theorem | ψ | W , ( D ′ ) ≤ C | ηψ | W , ( D ′ ) ≤ C | ψ | L ( D ) for all D ′ ⊂ D . In particular, this implies | ψ | L ∞ ( D ′ ) ≤ C | ψ | L ( D ) . At this point we may set p = 2 in (3.5) and find | ψ | W ,p ( D ′ ) ≤ C | ψ | W , ( D ′ ) ≤ C | ψ | L ( D ) , which proves the result. (cid:3) Remark 3.4.
In the case that λ = 0 the equation (2.4) arises from a conformally invariantenergy functional and is scale invariant. This scale invariance can be exploited to show thatsolutions of (2.4) cannot have isolated singularities, see [14], Theorem 3.1.By the main result of [3] we know that the nodal set of solutions to (2.4) on closed surfacesis discrete. The next Proposition gives an upper bound on their nodal set. VOLKER BRANDING
Proposition 3.5.
Suppose that ψ is a smooth solution of (2.4) that is not identically zero.Then the following inequality holds Z M ( λ + µ | ψ | ) d vol g ≥ πχ ( M ) + 4 πN, (3.6) where χ ( M ) is the Euler characteristic of the surface. Moreover, N denotes an estimate on thenodal set N = X p ∈ M, | ψ | ( p )=0 n p , where n p is the order of vanishing of | ψ | at the point p .Proof. By changing the connection on Σ M we obtain the following inequality (see [7], Lemma2.1 and references therein for a detailed derivation) h ψ, D ψ i| ψ | ≥ R | T | | ψ | − ∆ log | ψ | with the stress-energy tensor for the Dirac action T ( X, Y ) := h X · ∇ Y ψ + Y · ∇ X ψ, ψ i . Using(2.10) we find h ψ, D ψ i| ψ | = ( λ + µ | ψ | ) . We can estimate the stress-energy tensor as | T | ≥ λ + µ | ψ | ) , which yields ( λ + µ | ψ | ) ≥ K −
2∆ log | ψ | , where K = 2 R denotes the Gaussian curvature of M . By integrating over M and using thatfor a function with discrete zero set Z M ∆ log | ψ | d vol g = − π X p ∈ M, | ψ | ( p )=0 n p we obtain the result. (cid:3) Corollary 3.6. (1)
The estimate on the nodal set (3.6) generalizes the estimates on thenodal set for eigenspinors [7] and on solutions to non-linear Dirac equations [1] , Propo-sition 8.4. (2)
Due to the last Proposition we obtain the following upper bound on the nodal set ofsolutions to (2.4) N ≤ − χ ( M )2 + 14 π Z M ( λ + µ | ψ | ) d vol g . (3) We also obtain a vanishing result for surfaces of positive Euler characteristic: Moreprecisely, if Z M ( λ + µ | ψ | ) d vol g < π then we get a contradiction from (3.6) forcing ψ to be trivial. Using the Sobolev embedding theorem we can obtain another variant of the last statementfrom the previous Corollary.
Proposition 3.7.
Let ψ be a smooth solution of (2.4) . Suppose that there do not exist harmonicspinors on M . If | λ || ψ | L + | µ || ψ | L < ǫ (3.7) for some small ǫ > then ψ must be trivial. ONLINEAR DIRAC EQUATIONS, MONOTONICITY FORMULAS AND LIOUVILLE THEOREMS 9
Proof.
By assumption 0 is not in the spectrum of D and we can estimate | ψ | ≤ | λ | | Dψ | , where λ denotes the smallest eigenvalue of the Dirac operator. Making use of elliptic estimatesfor first order equations we find | ψ | L ≤ C | ψ | W , ≤ C ( | Dψ | L + | ψ | L ) ≤ C ( | λ || ψ | L + | µ || (cid:12)(cid:12) | ψ | (cid:12)(cid:12) L ) ≤ C ( | λ || ψ | L | ψ | L + | µ || ψ | L ) ≤ ǫC | ψ | L , where we made use of the assumptions. Thus, for ǫ small enough ψ has to vanish. (cid:3) The higher-dimensional case.Proposition 3.8.
Suppose that M is a closed Riemannian spin manifold with positive scalarcurvature. Suppose that ψ is a smooth solution of (2.4) with small energy, that is ( λ + µ | ψ | ) < R . (3.8) Then ψ vanishes identically.Proof. We use the Bochner formula (2.9) and calculate∆ 12 | ψ | = (cid:12)(cid:12) d | ψ | (cid:12)(cid:12) + | ψ | |∇ ψ | + | ψ | (cid:0) R − ( λ + µ | ψ | ) (cid:1) > | ψ | is a subharmonic function and due to the maximum principleit has to be constant. Thus, we obtain0 = | ψ | |∇ ψ | + | ψ | (cid:0) R − ( λ + µ | ψ | ) (cid:1) and the result follows by making use of the assumption. (cid:3) Nonlinear Dirac equations on complete manifolds
In this section we study the behavior of solutions of (2.4) on complete manifolds. We willderive several monotonicity formulas and, as an application, we obtain Liouville theorems.4.1.
A Liouville Theorem for stationary solutions.
In this section we will derive a van-ishing theorem for stationary solutions of (2.4).
Theorem 4.1.
Suppose that M = R n , H n with n ≥ . Let ψ ∈ L loc (Σ M ) × W , loc (Σ M ) be astationary solution of (2.4) . If λ ≥ , µ ≤ and Z M ( | ψ | + |∇ ψ | ) d vol g < ∞ (4.1) then ψ vanishes identically.Proof. We will first show the result for M = R n . Choose η ∈ C ∞ ( R ) such that η = 1 for r ≤ R , η = 0 for r ≥ R and | η ′ ( r ) | ≤ R . In addition, we choose Y ( x ) = xη ( r ) ∈ C ∞ ( M, R n ), where r = | x | . Then, we set k ij := ∂Y i ∂x j = η ( r ) δ ij + x i x j r η ′ ( r )and inserting this into the stationary condition (2.6) we obtain Z R n (2 h ψ, Dψ i − nµ | ψ | ) η ( r ) d vol g = − Z R n (2 h ψ, ∂ r · ∇ ∂ r ψ i − µ | ψ | ) rη ′ ( r ) d vol g . Using the equation for ψ we get Z R n (2 λ | ψ | + (cid:0) − n ) µ | ψ | ) η ( r ) d vol g = − Z R n (2 h ψ, ∂ r · ∇ ∂ r ψ i − µ | ψ | ) rη ′ ( r ) d vol g . The right hand side can be controlled as follows (cid:12)(cid:12) Z R n (2 h ψ, ∂ r · ∇ ∂ r ψ i − µ | ψ | ) rη ′ ( r ) d vol g (cid:12)(cid:12) ≤ C Z B R \ B R ( | ψ ||∇ ψ | + | ψ | ) dx. Making use of the assumptions on λ, µ and by the properties of the cut-off function η we obtain Z B R (2 λ | ψ | + (cid:0) − n ) µ | ψ | ) dx ≤ Z R n (2 λ | ψ | + (cid:0) − n ) µ | ψ | ) η ( r ) d vol g such that we get Z B R (2 λ | ψ | + (cid:0) − n ) µ | ψ | ) dx ≤ C Z B R \ B R ( | ψ ||∇ ψ | + | ψ | ) dx ≤ C Z B R \ B R ( |∇ ψ | + | ψ | ) dx. Taking the limit R → ∞ and making use of the finite energy assumption we obtain Z R n | ψ | (2 λ + (2 − n ) µ | ψ | ) d vol g ≤ , yielding the result. By applying the Theorem of Cartan-Hadamard the proof carries over tohyperbolic space. (cid:3) Remark 4.2.
In particular, the last Proposition applies in the case µ = 0, which correspondsto ψ being an eigenspinor with positive eigenvalue λ . Thus, there does not exist an eigenspinorsatisfying Z M ( | ψ | + |∇ ψ | ) d vol g < ∞ for a positive eigenvalue λ on M = R n , H n for n ≥ Monotonicity formulas for smooth solutions.
In this section we will derive a mono-tonicity formula for smooth solutions of (2.4) on complete Riemannian manifolds. We will makeuse of the fact that the stress-energy tensor (2.8) is divergence free, whenever ψ is a solution of(2.4). First of all, let us recall the following facts: A vector field X is called conformal if L X g = f g, where L denotes the Lie-derivative of the metric with respect to X and f : M → R is a smoothfunction. Lemma 4.3.
Let T be a symmetric 2-tensor. For any vector field X the following formulaholds div( ι X T ) = ι X div T + h T, ∇ X i . (4.2) If X is a conformal vector field, then the second term on the right hand side acquires the form h T, ∇ X i = 1 n div X tr T. (4.3)By integrating over a compact region U , making use of Stokes theorem, we obtain Lemma 4.4.
Let ( M, g ) be a Riemannian manifold and U ⊂ M be a compact region with smoothboundary. Then, for any symmetric -tensor and any vector field X the following formula holds Z ∂U T ( X, n ) dσ = Z U ι X div T dx + Z U h T, ∇ X i dx, (4.4) where n denotes the normal to U . The same formula holds for a conformal vector field X if wereplace the second term on the right hand by (4.3) . We now derive a type of monotonicity formula for smooth solutions of (2.4) in R n . ONLINEAR DIRAC EQUATIONS, MONOTONICITY FORMULAS AND LIOUVILLE THEOREMS 11
Proposition 4.5 (Monotonicity formula in R n ) . Let ψ be a smooth solution of (2.4) on M = R n . Let B R ( x ) be a geodesic ball around the point x ∈ M and < R < R ≤ R . Then thefollowing monotonicity formula holds R − n µ Z B R x | ψ | dx − R − n µ Z B R x | ψ | dx = − λ Z R R (cid:0) r − n Z B r ( x ) | ψ | dx (cid:1) dr (4.5)+ 2 Z R R (cid:0) r − n Z ∂B r ( x ) h ψ, ∂ r · ∇ ∂ r ψ i dσ (cid:1) dr. Proof.
For M = R n we choose the conformal vector field X = r ∂∂r with r = | x | . In this case wehave div( X ) = n , thus(2 − n ) µ Z B r | ψ | d vol g + rµ Z ∂B r | ψ | dσ = − λ Z B r | ψ | d vol g + 2 r Z ∂B r h ψ, ∂ r · ∇ ∂ r ψ i dσ, where we used (4.4) and (4.3). Making use of the coarea formula we can rewrite this as ddr (cid:0) r − n µ Z B r | ψ | d vol g (cid:1) = − λr − n Z B r | ψ | d vol g + 2 r − n Z ∂B r h ψ, ∂ r · ∇ ∂ r ψ i dσ and integrating with respect to r yields the result. (cid:3) Remark 4.6.
The previous monotonicity formula also holds if ψ was only a weak solution of(2.4), that is ψ ∈ L (Σ M ) × W , (Σ M ).We now aim at generalizing the monotonicity formula (4.5) to the case of a complete Rie-mannian spin manifold. Note that, in general, the vector field X = r ∂∂r will no be conformal.We fix a point x ∈ M and consider a ball with geodesic radius r = d ( x , · ) around that point,where d denotes the Riemannian distance function. Moreover, i M will refer to the injectivityradius of M . Using geodesic polar coordinates we decompose the metric in B i M with the helpof the Gauss Lemma as g = g r + dr ⊗ dr. In the following we will frequently make use of the Hessian of the Riemannian distance function.Since the Hessian is a symmetric bilinear form we may diagonalize it, its eigenvalues will bedenoted by ω i , i = 1 , . . . , n . Thus, we may write n X i =1 Hess( r )( e i , e i ) = n X i =1 ω i , where e i , i = 1 . . . n denotes an orthonormal basis of T M . We denote the largest eigenvalue by ω max and set Ω := 2 ω max − n X i =1 ω i . (4.6)The quantity Ω depends on the geometry of the manifold M and, in general, it cannot becomputed explicitly. For some explicit estimates on Ω in terms of the geometric data we referto [24], Lemma 3.2. In the case M = R n we have Ω = 2 − n . Proposition 4.7.
Let ( M, g ) be a complete Riemannian spin manifold and suppose that ψ isa smooth solution of (2.4) . Then for all < R < R ≤ R , R ∈ (0 , i M ) the following type ofmonotonicity formula holds R Ω1 µ Z B R | ψ | dx ≤ R Ω2 µ Z B R | ψ | dx + 2 λω max Z R R (cid:0) r Ω − Z B r | ψ | dx (cid:1) dr (4.7) − Z R R (cid:0) r Ω Z ∂B r h ψ, ∂ r · ∇ ∂ r ψ i dσ (cid:1) dr, where Ω is given by (4.6) . Proof.
Inserting the stress-energy tensor (2.8) into (4.4) and choosing the vector field X = r ∂∂r we obtain the following equation2 r Z ∂B r h ψ, ∂ r · ∇ ∂ r ψ i dσ − rµ Z ∂B r | ψ | dσ = Z B r h e i · ∇ e j ψ + e j · ∇ e i ψ, ψ i Hess( r )( e i , e j ) dx − µ Z B r tr Hess( r ) | ψ | dx. Diagonalizing the Hessian of the Riemannian distance function we find h e i · ∇ e j ψ + e j · ∇ e i ψ, ψ i Hess( r )( e i , e j ) − µ tr Hess( r ) | ψ | ≤ ω max h ψ, Dψ i − µ | ψ | n X i =1 ω i =2 ω max λ | ψ | + µ | ψ | Ω , where we used (4.6). Hence, we obtain µ Ω Z B r | ψ | dx + rµ Z ∂B r | ψ | dσ ≥ r Z ∂B r h ψ, ∂ r · ∇ ∂ r ψ i dσ − ω max Z B r λ | ψ | dx and by the coarea formula this yields ddr r Ω µ Z B r | ψ | dx ≥ − λω max r Ω − Z B r | ψ | dx + 2 r Ω Z ∂B r h ψ, ∂ r · ∇ ∂ r ψ i dσ. Integrating with respect to r then yields the claim. (cid:3) Remark 4.8.
The problematic contribution in the monotonicity formulas (4.5) and (4.8) is theindefinite term h ψ, ∂ r · ∇ ∂ r ψ i . To give this term a definite sign we could assume that ψ is botha solution of (2.4) and a twistor spinor. In this case we have h ψ, ∂ r · ∇ ∂ r ψ i = 1 n g ( ∂ r , ∂ r ) h ψ, Dψ i = 1 n g ( ∂ r , ∂ r )( λ | ψ | + µ | ψ | ) . The right hand side of this equation is positive for λ, µ >
0. However, we have already seen thatunder the assumptions from above | ψ | is equal to a constant and in this case the monotonicityformula contains no interesting information. Theorem 4.9 (Monotonicity formula) . Let ( M, g ) be a complete Riemannian spin manifoldand suppose that ψ is a smooth solution of (2.4) . Then for all < R < R ≤ R , R ∈ (0 , i M ) the following monotonicity type formula holds R Ω1 µ Z B R | ψ | dx ≤ CR Ω2 µ Z B R ( | ψ | + |∇ ψ | ) dx (4.8)+ C Z R R (cid:0) r Ω − Z B r ( | ψ | + | ψ | + |∇ ψ | ) dx (cid:1) dr, where Ω is given by (4.6) . The positive constant C only depends on the geometry of M .Proof. First of all, we manipulate the indefinite term from (4.7) using integration by parts Z R R (cid:0) r Ω Z ∂B r h ψ, ∂ r · ∇ ∂ r ψ i dσ (cid:1) dr = Z R R (cid:0) r Ω ddr Z B r h ψ, ∂ r · ∇ ∂ r ψ i dx (cid:1) dr = R Ω2 Z B R h ψ, ∂ r · ∇ ∂ r ψ i dx − R Ω1 Z B R h ψ, ∂ r · ∇ ∂ r ψ i dx − Ω Z R R (cid:0) r Ω − Z B r h ψ, ∂ r · ∇ ∂ r ψ i dx (cid:1) dr. Moreover, we have Z B r h ψ, ∂ r · ∇ ∂ r ψ i dx ≤ C Z B r | ψ | dx + C Z B r |∇ ψ | dx, which proves the result. (cid:3) ONLINEAR DIRAC EQUATIONS, MONOTONICITY FORMULAS AND LIOUVILLE THEOREMS 13
Remark 4.10.
To control the term involving the L norm of ∇ ψ we can also use ellipticestimates (localized to the geodesic ball B r ) for first order equations and find | ψ | W , ( B r ) ≤ C ( | Dψ | L ( B r ) + | ψ | L ( B r ) ) ≤ C ( | ψ | L ( B r ) + | ψ | L ( B r ) ) , where we used that ψ is a solution of (2.4).We can obtain a Liouville Theorem from (4.8) by assuming that R B R ( | ψ | + | ψ | + |∇ ψ | ) dx decays fast enough. However, these assumptions are rather restrictive. Theorem 4.11.
Let ( M, g ) be a complete Riemannian spin manifold and suppose that ψ is asmooth solution of (2.4) . If µ > and R Ω Z B R ( | ψ | + | ψ | + |∇ ψ | ) dx → , (4.9) as R → ∞ , then ψ vanishes identically.Proof. Suppose that 0 < r < R and fix a point x ∈ M . Inserting the decay assumptions into(4.8) we find that r Ω µ Z B r ( x | ψ | dx → R → ∞ yielding the result. (cid:3) A Liouville Theorems for complete manifolds with positive Ricci curvature.
In this section we will prove a Liouville theorem for smooth solutions of (2.4) on completenoncompact manifolds with positive Ricci curvature. Our result is motivated from a similarresult for harmonic maps, see [26], Theorem 1. We set e ( ψ ) := | ψ | . Theorem 4.12.
Let ( M, g ) be a complete noncompact Riemannian spin manifold with positiveRicci curvature. Suppose that Ric ≥ n ( λ + µ | ψ | ) g. (4.10) If ψ is a smooth solution of (2.4) with finite energy e ( ψ ) then ψ vanishes identically.Proof. Taking the trace of (4.10) we find( λ + µ | ψ | ) ≤ R e ( ψ ) ≥ (cid:12)(cid:12) d | ψ | (cid:12)(cid:12) . (4.11)In addition, by the Cauchy-Schwarz inequality we find | de ( ψ ) | ≤ e ( ψ ) (cid:12)(cid:12) d | ψ | (cid:12)(cid:12) . (4.12)We fix a positive number ǫ > p e ( ψ ) + ǫ = ∆ e ( ψ )2 p e ( ψ ) + ǫ − | de ( ψ ) | ( e ( ψ ) + ǫ ) ≥ (cid:12)(cid:12) d | ψ | (cid:12)(cid:12) p e ( ψ ) + ǫ (cid:0) − e ( ψ ) e ( ψ ) + ǫ (cid:1) ≥ , where we used (4.11) and (4.12). Let η be an arbitrary function on M with compact support.We obtain0 ≤ Z M η p e ( ψ ) + ǫ ∆ p e ( ψ ) + ǫd vol g = − Z M η p e ( ψ ) + ǫ h∇ η, ∇ p e ( ψ ) + ǫ i d vol g − Z M η |∇ p e ( ψ ) + ǫ | d vol g . Now let x be a point in M and let B R , B R be geodesic balls centered at x with radii R and2 R . We choose a cutoff function η satisfying η ( x ) = ( , x ∈ B R , , x ∈ M \ B R . In addition, we choose η such that 0 ≤ η ≤ , |∇ η | ≤ CR for a positive constant C . Then, we find0 ≤ − Z B R η p e ( ψ ) + ǫ h∇ η, ∇ p e ( ψ ) + ǫ i dx − Z B R η |∇ p e ( ψ ) + ǫ | dx ≤ (cid:0) Z B R \ BR η | p e ( ψ ) + ǫ | dx (cid:1) (cid:0) Z B R \ BR |∇ η | ( e ( ψ ) + ǫ ) dx (cid:1) − Z B R \ B R η |∇ p e ( ψ ) + ǫ | dx − Z B R |∇ p e ( ψ ) + ǫ | dx. We therefore obtain Z B r |∇ p e ( ψ ) + ǫ | dx ≤ Z B R \ B R |∇ η | ( e ( ψ ) + ǫ ) dx ≤ C R Z B R ( e ( ψ ) + ǫ ) dx. We set B ′ R := B R \ { x ∈ B R | e ( ψ )( x ) = 0 } and find Z B ′ r |∇ ( e ( ψ ) + ǫ ) | e ( ψ ) + ǫ ) dx ≤ C R Z B R ( e ( ψ ) + ǫ ) dx. Letting ǫ → Z B ′ r |∇ ( e ( ψ ) | e ( ψ ) dx ≤ C R Z B R e ( ψ ) dx. Now, letting R → ∞ and under the assumption that the energy is finite, we have Z M \{ e ( ψ )=0 } |∇ e ( ψ ) | e ( ψ ) d vol g ≤ , hence the energy e ( ψ ) has to be constant. If e ( ψ ) = 0, then the volume of M would have to befinite. However, by Theorem 7 of [31] the volume of a complete and noncompact Riemannianmanifold with nonnegative Ricci curvature is infinite. Hence e ( ψ ) = 0, which yields the result. (cid:3) Note, that Theorem 4.12 also holds in the case µ = 0, which gives us the following vanishingresult for eigenspinors: Corollary 4.13.
Suppose that ψ is a smooth solution of Dψ = λψ on a complete noncompactmanifold with positive Ricci curvature. If Ric ≥ n λ g and e ( ψ ) is finite then ψ vanishes identically. Dirac-harmonic maps with curvature term from complete manifolds
Dirac-harmonic maps with curvature term arise as critical points of part of the supersym-metric nonlinear σ -model from quantum field theory [15], p. 268. They form a pair of a mapfrom a Riemann spin manifold to another Riemannian manifold coupled with a vector spinor.For a two-dimensional domain Dirac-harmonic they belong to the class of conformally invariantvariational problems. The conformal invariance gives rise to a removable singularity theorem[6] and an energy identity [32]. ONLINEAR DIRAC EQUATIONS, MONOTONICITY FORMULAS AND LIOUVILLE THEOREMS 15
The mathematical study of the supersymmetric nonlinear σ -model was initiated in [12], wherethe notion of Dirac-harmonic maps was introduced. The full action of the supersymmetricnonlinear σ -model contains two additional terms: Taking into account and additional two-formin the action functional the resulting equations were studied in [8], Dirac-harmonic maps withcurvature term to target spaces with torsion are analyzed in [10]. However, most of the resultspresented in this section still hold true if we would consider the full supersymmetric nonlinear σ -model.In the following we still assume that ( M, g ) is a complete Riemannian spin manifold and (
N, h )another Riemannian manifold. Whenever we will make use of indices we use Latin letters forindices related to M and Greek letters for indices related to N . Let φ : M → N be a mapand let φ ∗ T N be the pull-back of the tangent bundle from N . We consider the twisted bundleΣ M ⊗ φ ∗ T N , on this bundle we obtain a connection induced from Σ M and φ ∗ T N , which willbe denoted by ˜ ∇ . Section in Σ M ⊗ φ ∗ T N are called vector spinors . On Σ M ⊗ φ ∗ T N we have ascalar product induced from Σ M and φ ∗ T N , we will denote its real part by h· , ·i . The twistedDirac operator acting on vector spinors is defined as /D := n X i =1 e i · ˜ ∇ e i . Note that the operator /D is still elliptic. Moreover, we assume that the connection on φ ∗ T N is metric, thus /D is also self-adjoint with respect to the L -norm if M is compact. The energyfunctional for Dirac-harmonic maps with curvature term is given by E c ( φ, ψ ) = 12 Z M ( | dφ | + h ψ, /Dψ i − h R N ( ψ, ψ ) ψ, ψ i ) d vol g . (5.1)Here, R N denotes the curvature tensor of the manifold N . The factor 1 / h R N ( ψ, ψ ) ψ, ψ i = R αβγδ h ψ α , ψ γ ih ψ β , ψ δ i , which ensures that the functional is real valued. The critical points of the energy functional(5.1) are given by τ ( φ ) = 12 R N ( ψ, e α · ψ ) dφ ( e α ) − h ( ∇ R N ) ♯ ( ψ, ψ ) ψ, ψ i , (5.2) /Dψ = 13 R N ( ψ, ψ ) ψ, (5.3)where τ ( φ ) ∈ Γ( φ ∗ T N ) is the tension field of the map φ and ♯ : φ ∗ T ∗ N → φ ∗ T N represents themusical isomorphism. For a derivation see [11], Section II and [9], Proposition 2.1.Solutions ( φ, ψ ) of the system (5.2), (5.3) are called
Dirac-harmonic maps with curvatureterm .The correct function space for weak solutions of (5.2), (5.3) is χ ( M, N ) := W , ( M, N ) × W , ( M, Σ M ⊗ φ ∗ T N ) × L ( M, Σ M ⊗ φ ∗ T N ) . For the domain being a closed surface it was shown in [9] that a weak solution ( φ, ψ ) ∈ χ ( M, N )of (5.2), (5.3) is smooth.
Definition 5.1.
A weak Dirac-harmonic map with curvature term ( φ, ψ ) ∈ χ ( M, N ) is called stationary if it is also a critical point of E c ( φ, ψ ) with respect to domain variations.To obtain the formula for stationary Dirac-harmonic maps with curvature term we make useof the same methods as before. Since the twist bundle φ ∗ T N does not depend on the metric on M we can use the same methods as in Section 2. Thus, let k be a smooth element of Sym(0 , ψ g + tk := β g,g + tk ψ ∈ Γ(Σ M g + tk ⊗ φ ∗ T N ). Lemma 5.2.
The following formula for the variation of the twisted Dirac-energy with respectto the metric holds ddt (cid:12)(cid:12) t =0 h ψ g + tk , /D g + tk ψ g + tk i Σ g + tk M ⊗ φ ∗ T N (5.4)= − h e i · ∇ Σ g M ⊗ φ ∗ T Ne j ψ + e j · ∇ Σ g M ⊗ φ ∗ T Ne i ψ, ψ i Σ g M ⊗ φ ∗ T N k ij with the stress-energy tensor associated to the twisted Dirac energy on the right hand side. At this point we are ready to compute the variation of the energy functional (5.1) with respectto the metric.
Proposition 5.3.
Let the pair ( φ, ψ ) ∈ χ ( M, N ) be a weak Dirac-harmonic map with curvatureterm. Then ( φ, ψ ) is a stationary Dirac-harmonic map with curvature term if for any smooth (0 , -tensor k the following formula holds Z M (cid:0) h dφ ( e i ) , dφ ( e j ) i − g ij | dφ | + 12 h ψ, e i · ∇ Σ g M ⊗ φ ∗ T Ne j ψ + e j · ∇ Σ g M ⊗ φ ∗ T Ne i ψ i (5.5) − g ij h R N ( ψ, ψ ) ψ, ψ i (cid:1) k ij ) d vol g = 0 . Proof.
We calculate ddt (cid:12)(cid:12) t =0 E c ( φ, ψ, g + tk ) = 0 , where k is a symmetric (0 , t some small number. Using the variation of thevolume-element (2.7) we obtain the variation of the Dirichlet energy ddt (cid:12)(cid:12) t =0 Z M | dφ | g + tk d vol g + tk = Z M (cid:0) − h h ( dφ ( e i ) , dφ ( e j )) , k ij i d vol g + 12 | dφ | h g, k i g d vol g (cid:1) . Note that we get a minus sign in the first term since dφ ∈ Γ( T ∗ M ⊗ φ ∗ T N ) such that we haveto vary the metric on the cotangent bundle. As a second step, we compute the variation of theDirac energy using (5.4) and (2.7) yielding ddt (cid:12)(cid:12) t =0 Z M h ψ g + tk , /D g + tk ψ g + tk i d vol g + tk = Z M − h e i · ∇ Σ g M ⊗ φ ∗ T Ne j ψ + e j · ∇ Σ g M ⊗ φ ∗ T Ne i ψ, ψ i Σ g M k ij d vol g + 12 Z M h ψ, /Dψ ih g, k i g d vol g . Finally, for the term involving the curvature tensor of the target and the four spinors we obtain ddt (cid:12)(cid:12) t =0 Z M h R N ( ψ g + tk , ψ g + tk ) ψ g + tk ,ψ g + tk i g + tk d vol g + tk = ddt (cid:12)(cid:12) t =0 Z M h R N ( ψ, ψ ) ψ, ψ i Σ g + tk M ⊗ φ ∗ T N d vol g + tk = 12 Z M h R N ( ψ, ψ ) ψ, ψ i Σ g M ⊗ φ ∗ T N h g, k i g d vol g , where we used that β acts as an isometry on the spinor bundle in the first step. Adding up thethree contributions and using the fact that ( φ, ψ ) is a weak Dirac-harmonic map with curvatureterm yields the result. (cid:3) A Liouville Theorem for stationary solutions.
It is well known that a stationaryharmonic map R q → N with finite Dirichlet energy is a constant map [16], Section 5. Thisresult was generalized to stationary Dirac-harmonic maps and here we generalize it to stationaryDirac-harmonic maps with curvature term by adding a curvature assumption. A similar resultfor smooth Dirac-harmonic maps with curvature term was already obtained in [11], Theorem1.2. ONLINEAR DIRAC EQUATIONS, MONOTONICITY FORMULAS AND LIOUVILLE THEOREMS 17
Theorem 5.4.
Let M = R n , H n with dim M ≥ and suppose that ( φ, ψ ) ∈ W , loc ( M, N ) × W , loc ( M, Σ M ⊗ φ ∗ T N ) × L loc ( M, Σ M ⊗ φ ∗ T N ) is a stationary Dirac-harmonic maps with cur-vature term satisfying Z R n ( | dφ | + |∇ Σ M ψ | + | ψ | ) d vol g < ∞ . (5.6) If N has positive sectional curvature then φ is constant and ψ vanishes identically.Proof. Let η ∈ C ∞ ( R ) be a smooth cut-off function satisfying η = 1 for r ≤ R , η = 0 for r ≥ R and | η ′ ( r ) | ≤ CR . In addition, we choose Y ( x ) := xη ( r ) ∈ C ∞ ( R n , R n ) with r = | x | . Hence, wefind k ij = ∂Y i ∂x j = δ ij η ( r ) + x i x j r η ′ ( r ) . Inserting this into (5.5) and using that ( φ, ψ ) is a weak solution of the system (5.2), (5.3) weobtain(2 − n ) Z R n ( | dφ | + 16 h R N ( ψ, ψ ) ψ, ψ i ) η ( r ) d vol g = Z R n ( | dφ | − (cid:12)(cid:12) ∂φ∂r (cid:12)(cid:12) − h ψ, ∂ r · ˜ ∇ ∂ r ψ i + 16 h R N ( ψ, ψ ) ψ, ψ i ) rη ′ ( r ) d vol g . By the properties of the cut-off function η we find (see the proof of Theorem 4.1 for more details)(2 − n ) Z R n ( | dφ | + 16 h R N ( ψ, ψ ) ψ, ψ i ) η ( r ) d vol g ≤ C Z B R \ B R ( | dφ | + | ψ || ˜ ∇ ψ | + | ψ | ) dx ≤ C Z B R \ B R ( | dφ | + |∇ Σ M ψ | + | ψ | ) dx. Due to the finite energy assumption and the fact that n ≥
3, taking the limit R → ∞ yields Z R n ( | dφ | + 16 h R N ( ψ, ψ ) ψ, ψ i ) d vol g = 0 . The statement follows since N has positive sectional curvature, see [11], p.15 for more details.To obtain the result for hyperbolic space we again apply the theorem of Cartan-Hadamard. (cid:3) Monotonicity formulas and Liouville Theorems.
In this section we derive a mono-tonicity formula for Dirac-harmonic maps with curvature term building on their stress-energytensor. For simplicity, we will mostly assume that ( φ, ψ ) is a smooth Dirac-harmonic map withcurvature term. From (5.5) we obtain the stress-energy tensor for the functional E c ( φ, ψ ) as S ij =2 h dφ ( e i ) , dφ ( e j ) i − g ij | dφ | (5.7)+ 12 h ψ, e i · ∇ Σ M ⊗ φ ∗ T Ne j ψ + e j · ∇ Σ M ⊗ φ ∗ T Ne i ψ i − g ij h R N ( ψ, ψ ) ψ, ψ i . It was shown in [9], Proposition 3.2, that the stress-energy tensor is divergence free when ( φ, ψ ) isa smooth Dirac-harmonic map with curvature term. For a Dirac-harmonic map with curvatureterm the trace of (5.7) can easily be computed and gives g ij S ij = (2 − n )( | dφ | + 16 h R N ( ψ, ψ ) ψ, ψ i ) . Hence, we will consider the following energy e c ( φ, ψ ) := | dφ | + 16 h R N ( ψ, ψ ) ψ, ψ i and study its monotonicity. Note that e c ( φ, ψ ) ≥ N has positive sectional curvature.Moreover, since for a solution of (5.3) h ψ, /Dψ i = 13 h R N ( ψ, ψ ) ψ, ψ i the assumption that N has positive sectional curvature forces /D to have a positive spectrum. Proposition 5.5 (Monotonicity formula in R n ) . Let ( φ, ψ ) be a smooth solution of (5.2) , (5.3) for M = R n . Let B R ( x ) be a geodesic ball around the point x ∈ M and < R < R ≤ R .Then the following following monotonicity formula holds R − n Z B R e c ( φ, ψ ) dx = R − n Z B R e c ( φ, ψ ) dx (5.8) − Z R R (cid:0) r − n Z ∂B r (2 (cid:12)(cid:12) ∂φ∂r (cid:12)(cid:12) + h ψ, ∂ r · ˜ ∇ ∂ r ψ i ) dσ (cid:1) dr. Proof.
For M = R n we choose the conformal vector field X = r ∂∂r with r = | x | . In this case wehave div( X ) = n , thus we obtain r Z ∂B r ( x ) (2 (cid:12)(cid:12) ∂φ∂r (cid:12)(cid:12) + h ψ, ∂ r · ˜ ∇ ∂ r ψ i − e c ( φ, ψ )) dx = (2 − n ) Z B r ( x ) ( | dφ | + 16 h R N ( ψ, ψ ) ψ, ψ i ) dx, where we used (4.4) and (4.3). This can be rewritten as(2 − n ) Z B r ( x ) e c ( φ, ψ ) + r Z ∂B r ( x ) e c ( φ, ψ ) = r Z ∂B r ( x ) (2 (cid:12)(cid:12) ∂φ∂r (cid:12)(cid:12) + h ψ, ∂ r · ˜ ∇ ∂ r ψ i ) dx and by applying the coarea formula we find ddr (cid:0) r − n Z B r e c ( φ, ψ ) dx (cid:1) = r − n Z ∂B r (2 (cid:12)(cid:12) ∂φ∂r (cid:12)(cid:12) + h ψ, ∂ r · ˜ ∇ ∂ r ψ i ) dσ. The result then follows by integration with respect to r . (cid:3) Remark 5.6.
The last statement also holds if ( φ, ψ ) is a weak Dirac-harmonic map withcurvature term, that is ( φ, ψ ) ∈ χ ( M, N ) for M = R n . It this case we can require higherintegrability assumptions on ψ as in [30], Proposition 4.5 to get the following result: Let thepair ( φ, ψ ) be a weak Dirac-harmonic map with curvature term in some domain D ⊂ R n . Inaddition, suppose that ∇ ψ ∈ L p ( D ) for n < p ≤ n , then R − n Z B R e c ( φ, ψ ) dx ≤ R − n Z B R e c ( φ, ψ ) dx + C R − np . Here, the constant C only depends on |∇ ψ | L p ( D ) .A possible application of this monotonicity formula for stationary Dirac-harmonic maps withcurvature term is to calculate the Hausdorff dimension of their singular set. For Dirac-harmonicmaps this has been carried out in [30], Proposition 4.5.To derive a monotonicity formula on a Riemannian manifold we again fix a point x ∈ M and consider a ball with geodesic radius r = d ( x , · ) around that point, where d denotes theRiemannian distance function. Proposition 5.7.
Let ( φ, ψ ) be a smooth solution of the system (5.2) , (5.3) . Then for all < R < R ≤ R , R ∈ (0 , i M ) the following monotonicity type formula holds R Ω1 Z B R e c ( φ, ψ ) dx ≤ R Ω2 Z B R e c ( φ, ψ ) dx − Z R R r Ω Z ∂B r (cid:0) (cid:12)(cid:12) ∂φ∂r (cid:12)(cid:12) + h ψ, ∂ r · ˜ ∇ ∂ r ψ i (cid:1) dσ. (5.9) Proof.
We apply (4.4), therefore we calculate h S ij , Hess( r )( e i , e j ) i = − tr Hess( r ) e c ( φ, ψ )+ (cid:0) h dφ ( e i ) , dφ ( e j ) i + h ψ, e i · ˜ ∇ e j ψ + e j · ˜ ∇ e i ψ i (cid:1) Hess( r )( e i , e j ) . Moreover, we again make use of the quantity Ω defined in (4.6). By a calculation similar to onein the proof of Proposition 4.8 we obtain r Z ∂B r e c ( φ, ψ ) dx + Ω Z B r e c ( φ, ψ ) dσ ≥ r Z ∂B r (cid:0) (cid:12)(cid:12) ∂φ∂r (cid:12)(cid:12) + h ψ, ∂ r · ˜ ∇ ∂ r ψ i (cid:1) dσ. ONLINEAR DIRAC EQUATIONS, MONOTONICITY FORMULAS AND LIOUVILLE THEOREMS 19
Again, making use of the coarea formula this yields ddr r Ω Z B r e c ( φ, ψ ) dx ≥ r Ω Z ∂B r (cid:0) (cid:12)(cid:12) ∂φ∂r (cid:12)(cid:12) + h ψ, ∂ r · ˜ ∇ ∂ r ψ i (cid:1) dσ and the statement follows by integration with respect to r . (cid:3) Again, the presence of the Dirac-Term on the right hand side of (5.9) is an obstacle to amonotonicity formula. We can try to improve the result if we assume that the solution ψ of(5.3) has some additional structure. Definition 5.8.
We call ψ ∈ Γ(Σ M ⊗ φ ∗ T N ) a vector twistor spinor if it satisfies˜ ∇ X ψ + 1 n X · /Dψ = 0 (5.10)for all vector fields X . Remark 5.9.
If we assume that ψ is both a vector twistor spinor and a solution of (5.3) wefind ˜ ∇ X ψ = − n R N ( ψ, ψ ) X · ψ, for all vector fields X . Moreover, a direct calculation yields ∂ X | ψ | = h ˜ ∇ X ψ, ψ i = − n h R N ( ψ, ψ ) X · ψ, ψ i = − n R αβγδ h ψ α , ψ δ ih ψ β , X · ψ γ i . On the other hand we find R αβγδ h ψ α , ψ δ ih ψ β , X · ψ γ i = R αβγδ h ψ δ , ψ α ih X · ψ γ , ψ β i = − R αβγδ h ψ α , ψ δ ih ψ β , X · ψ γ i . Consequently the above expression is both purely imaginary and also purely real and thus hasto vanish, meaning that | ψ | has constant norm. Thus, this approach does not lead to aninteresting monotonicity formula. Theorem 5.10.
Let ( φ, ψ ) be a smooth solution of the system (5.2) , (5.3) . Then for all There is another way to bound R B r |∇ Σ M ψ | dx . Using the local form of theequation for the vector spinor (5.3) we obtain | Dψ | ≤ C ( | dφ || ψ | + | ψ | ) , where D denotes the classical Dirac operator. Via elliptic estimates this yields |∇ Σ M ψ | L ≤ C ( | dφ | L + | ψ | L + | ψ | L ) . Theorem 5.12. Let ( φ, ψ ) be a smooth solution of the system (5.2) , (5.3) and suppose that N has positive sectional curvature. If R Ω Z B R ( | dφ | + | ψ | + |∇ Σ M ψ | ) dx → as R → ∞ , then φ maps to a point and ψ is a trivial.Proof. This follows directly from (5.11) and the assumption. (cid:3) A Liouville Theorem for a domain with positive Ricci curvature. In this sectionwe derive a vanishing theorem for Dirac-harmonic maps with curvature term under an energyand curvature assumption, similar to Theorem 4.12. To this end we set e ( φ, ψ ) := 12 ( | dφ | + | ψ | ) . Lemma 5.13 (Bochner formulas) . Let ( φ, ψ ) be a smooth solution of the system (5.2) , (5.3) .Then the following Bochner formulas hold ∆ 12 | ψ | = (cid:12)(cid:12) d | ψ | (cid:12)(cid:12) + 2 | ψ | | ˜ ∇ ψ | + 29 | ψ | | R N ( ψ, ψ ) ψ | + R | ψ | (5.13)+ 12 | ψ | h e i · e j · R N ( dφ ( e i ) , dφ ( e j )) ψ, ψ i , ∆ 12 | dφ | = |∇ dφ | + h dφ ( Ric M ( e i )) , dφ ( e i ) i − h R N ( dφ ( e i ) , dφ ( e j )) dφ ( e i ) , dφ ( e j ) i (5.14)+ 12 h ( ∇ dφ ( e j ) R N )( e i · ψ, ψ ) dφ ( e i ) , dφ ( e j ) i + h R N ( e i · ψ, ˜ ∇ e j ψ ) dφ ( e i ) , dφ ( e j ) i + 12 h R N ( e i · ψ, ψ ) ∇ e j dφ ( e i ) , dφ ( e j ) i + 112 hh ( ∇ dφ ( e i ) ( ∇ R N ) ♯ )( ψ, ψ ) ψ, ψ i , dφ ( e i ) i + 13 hh ( ∇ R N ) ♯ ( ˜ ∇ e i ψ, ψ ) ψ, ψ i , dφ ( e i ) i , where e i , i = 1 . . . , n is an orthonormal basis of T M .Proof. We choose a local orthonormal basis of T M such that ∇ e i e j = 0 , i, j = 1 , . . . , n atthe considered point. The fist equation follows by a direct calculation using the Weitzenb¨ockformula for the twisted Dirac-operator /D , that is /D ψ = − ˜∆ ψ + 14 Rψ + 12 e i · e j · R N ( dφ ( e i ) , dφ ( e j )) ψ, where ˜∆ denotes the connection Laplacian on the vector bundle Σ M ⊗ φ ∗ T N . To obtain thesecond equation we recall the following Bochner formula for a map φ : M → N ∆ 12 | dφ | = |∇ dφ | + h dφ (Ric M ( e i )) , dφ ( e i ) i − h R N ( dφ ( e i ) , dφ ( e j )) dφ ( e i ) , dφ ( e j ) i + h∇ τ ( φ ) , dφ i . Moreover, by a direct calculation we obtain˜ ∇ e j (cid:0) R N ( e i · ψ, ψ ) dφ ( e i ) (cid:1) = 12 ( ∇ dφ ( e j ) R N )( e i · ψ, ψ ) dφ ( e i ) + R N ( e i · ψ, ˜ ∇ e j ψ ) dφ ( e i )+ 12 R N ( e i · ψ, ψ ) ∇ e j dφ ( e i ) , ˜ ∇ e j (cid:0) h ( ∇ R N ) ♯ ( ψ, ψ ) ψ, ψ i (cid:1) = 112 h ( ∇ dφ ( e j ) ( ∇ R N ) ♯ )( ψ, ψ ) ψ, ψ i + 13 h ( ∇ R N ) ♯ ( ˜ ∇ e j ψ, ψ ) ψ, ψ i , which concludes the proof. (cid:3) ONLINEAR DIRAC EQUATIONS, MONOTONICITY FORMULAS AND LIOUVILLE THEOREMS 21 Corollary 5.14. Let ( φ, ψ ) be a smooth solution of the system (5.2) , (5.3) . Then the followingestimate holds: ∆ e ( φ, ψ ) ≥ c ( |∇ dφ | + (cid:12)(cid:12) d | ψ | (cid:12)(cid:12) ) − c e ( φ, ψ ) − c ( e ( φ, ψ )) , (5.15) where c i , i = 1 , , are positive constants that depend only on the geometry of M and N .Proof. Making use of the Bochner formulas we find∆ e ( φ, ψ ) ≥|∇ dφ | + κ M | dφ | + κ N | dφ | − |∇ R N | L ∞ √ n | ψ | | dφ | − | R N | L ∞ √ n | ψ || ˜ ∇ ψ || dφ | − | R N | L ∞ √ n | ψ | |∇ dφ || dφ | − |∇ R N | L ∞ | ψ | | dφ | − |∇ R N | L ∞ | ˜ ∇ ψ || ψ | | dφ | + (cid:12)(cid:12) d | ψ | (cid:12)(cid:12) + 2 | ψ | | ˜ ∇ ψ | + 29 | ψ | | R N ( ψ, ψ ) ψ | + R | ψ | − n | R N | L ∞ | ψ | | dφ | , where κ M denotes a lower bound for the Ricci curvature of M and κ N an upper bound for thesectional curvature of N . By application of Young’s inequality we find∆ e ( φ, ψ ) ≥ (1 − δ ) |∇ dφ | + | ψ | | ˜ ∇ ψ | (2 − δ − δ ) + (cid:12)(cid:12) d | ψ | (cid:12)(cid:12) (5.16)+ 29 | ψ | | R N ( ψ, ψ ) ψ | + R | ψ | + κ M | dφ | − | dφ | ( − κ N + 1 δ n | R N | L ∞ + δ ) − | ψ | | dφ | (cid:0) n δ | R N | L ∞ + 1 δ |∇ R N | L ∞ 36 + |∇ R N | L ∞ 12 + n | R N | L ∞ n δ |∇ R N | L ∞ (cid:1) with positive constants δ i , i = 1 , . . . 4. The statement then follows by applying Young’s inequal-ity again. (cid:3) Remark 5.15. (1) The analytic structure of (5.15) is the same as in the case of harmonicmaps.(2) If we want to derive a Liouville Theorem from (5.15) making only assumptions on thegeometry of M and N we would require that both c ≤ c ≤ 0. However, it caneasily be checked that we cannot achieve such an estimate since the curvature tensor of N appears on the right hand side of the system (5.2) and (5.3).However, we can give a Liouville theorem under similar assumptions as in Theorem 4.12. Asimilar Theorem for Dirac-harmonic maps was obtained in [13], Theorem 4. Theorem 5.16. Let ( M, g ) be a complete noncompact Riemannian spin manifold and ( N, h ) be a Riemannian manifold with nonpositive curvature. Suppose that ( φ, ψ ) is a Dirac-harmonicmap with curvature term with finite energy e ( φ, ψ ) . If Ric M ≥ ( c | ψ | + c | dφ | ) g, (5.17) with the constants c = n | R N | L ∞ + n | R N | L ∞ + (cid:0) 136 + n (cid:1) |∇ R N | L ∞ + |∇ R N | L ∞ ,c = n | R N | L ∞ + 1 then φ maps to a point and ψ vanishes identically.Proof. First of all we note that | de ( φ, ψ ) | = | d ( | dφ | + | ψ | ) | ≤ ( | dφ | |∇ dφ | + | ψ | (cid:12)(cid:12) d | ψ | (cid:12)(cid:12) + 2 | dφ ||∇ dφ || ψ | (cid:12)(cid:12) d | ψ | (cid:12)(cid:12) ) (5.18) ≤ e ( φ, ψ )( |∇ dφ | + (cid:12)(cid:12) d | ψ | (cid:12)(cid:12) ) . If we put δ = , δ = δ = δ = 1 in (5.16) we find∆ e ( φ, ψ ) ≥ 12 ( |∇ dφ | + (cid:12)(cid:12) d | ψ | (cid:12)(cid:12) )+ | dφ | (cid:18) Ric M −| ψ | (cid:0) n | R N | L ∞ + n | R N | L ∞ + (cid:0) 136 + n (cid:1) |∇ R N | L ∞ + |∇ R N | L ∞ (cid:1) g + | dφ | (cid:0) n | R N | L ∞ (cid:1) g (cid:19) Making use of the assumption (5.17) this yields∆ e ( φ, ψ ) ≥ δ ( |∇ dφ | + (cid:12)(cid:12) d | ψ | (cid:12)(cid:12) ) (5.19)for a positive constant δ . We fix a positive number ǫ > p δe ( φ, ψ ) + ǫ = δ ∆ e ( φ, ψ )2 p e ( φ, ψ ) + ǫ − δ |∇ e ( φ, ψ ) | ( e ( φ, ψ ) + ǫ ) ≥ δ |∇ dφ | + (cid:12)(cid:12) d | ψ | (cid:12)(cid:12) p e ( φ, ψ ) + ǫ (cid:0) − e ( φ, ψ ) e ( φ, ψ ) + ǫ (cid:1) ≥ , where we used (5.18) and (5.19). The rest of the proof is identical to the proof of Theorem4.12. (cid:3) References [1] Bernd Ammann. The smallest Dirac eigenvalue in a spin-conformal class and cmc immersions. Comm. Anal.Geom. , 17(3):429–479, 2009.[2] Christian B¨ar. 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