Fredholm properties of the $L^{2}$ exponential map on the symplectomorphism group

Let $M$ be a closed symplectic manifold with compatible symplectic form and Riemannian metric $g$. Here it is shown that the exponential mapping of the weak $L^{2}$ metric on the group of symplectic diffeomorphisms of $M$ is a non-linear Fredholm map of index zero. The result provides an interesting contrast between the $L^{2}$ metric and Hofer's metric as well as an intriguing difference between the $L^{2}$ geometry of the symplectic diffeomorphism group and the volume-preserving diffeomorphism group.


1.
Introduction. Let M be a closed symplectic manifold with symplectic form ω and Riemannian metric g. We assume ω and g are compatible in the sense that there exists an almost complex structure (an endomorphism of the tangent bundle T M ) satisfying J 2 = −I and ω(v, Jw) = g(v, w). The volume form defined by ω coincides with the volume form supplied by the metric g and we denote this form by µ. Let D s ω (M ) denote the group of all diffeomorphisms of Sobolev class H s preserving the Symplectic form ω on M . When s > dim M 2 + 1, D s ω (M ) becomes an infinite dimensional manifold whose tangent space at the identity T e D s ω (M ) is given by H s vector fields on M satisfying L v ω = 0. Using right translations, the L 2 inner-product defines a weak, right-invariant Riemannian metric on D s ω ; see the paper of Ebin and Marsden [5] for basic facts regarding D s ω (M ) . The action of D s ω (M ) on D s (M ) by composition on the right is an isometry of (1) and combined with the Hodge decomposition gives an L 2 orthogonal splitting of each tangent space where ω : T M → T * M is an isomorphism defined by X → i X ω with inverse ω : T * M → T M given by contracting a 1-form with the inverse components of the Symplectic form. The projections onto the first and second summands of (2) will be denoted by P η and Q η , respectively, or simply by P and Q if η = e the identity. By right-invariance P η (X) = dR η • P • dR η −1 (X), for X ∈ T η D s . Diffeomorphism groups can be realized as the configuration spaces of a number of equations in mathematical physics, and this provides a strong motivation to study their geometry. Perhaps the most famous example is the Euler equations of hydrodynamics, where Arnold, [1], noticed that a curve η(t) in the group of smooth volume preserving diffeomorphisms is a geodesic of the L 2 metric (1) if and only if the vector field v, defined by ∂ t η = v • η, solves the Euler equations of hydrodynamics.
Analogously, a curve η(t) in D s ω (M ) is a geodesic of the L 2 metric (1) starting from the identity in the direction v o if and only if the time dependent vector field v =η • η −1 on M solves the Symplectic Euler equations The geodesic equation on D s ω , corresponding to the Symplectic Euler equations, is a smooth ODE which can be solved for small values of t, cf. [4]. Furthermore, since the solutions depend smoothly on initial data, it follows that the L 2 metric has a smooth exponential map defined, for small t, by where η is the unique geodesic from the identity with initial velocity v o ∈ T e D s ω . It is well known that if M is a closed surface then solutions to the Euler equations of hydrodynamics exist globally in time. However, when M is three dimensional the existence of global solutions to the Euler equations is a celebrated open problem. In contrast, if M is a closed Symplectic manifold of any dimension then solutions to the Symplectic Euler equations exist globally in time and the map (4) is defined on the whole tangent space T e D s ω , cf. [4]. The subgroup of Hamiltonian diffeomorphisms plays a role in plasma dynamics analogous to the role played by the volume preserving diffeomorphism group in incompressible hydrodynamics, [2], [7], [16], [10], [11], [20]. Given a Symplectic manifold M of dimension 2n, the Vlasov Hamiltonian is defined by where the kernel G is taken so that it defines an appropriate norm on the space of densities of M . When G = (− ) −1 , the geodesic Vlasov equation, obtained via H, is given by the Symplectic Euler equations (3) rewritten on the space of Hamiltonian functions: where the time-dependent Hamiltonian F is related to the solution v of the Symplectic Euler equations by v = J∇F , and {·, ·} is the Poisson bracket defined by ω. We note that this is also the 2D Helmholtz equation of incompressible fluids, cf.
[23], [7]. A bounded linear operator S between Banach spaces is said to be F redholm if it has closed range, finite dimensional Kernel and finite dimensional co-kernel. S is said to be semi-Fredholm if it has closed range and at least one of the other two conditions holds. The index of a semi-Fredholm operator is defined as ind S = dim ker S − dim co ker S, and is a continuous function on the set of Fredholm operators into Z ∪ {±∞}. A smooth map f : M → N between Banach manifolds is called a Fredholm map if its Frechet derivative df (p) is a Fredholm operator for each p. If the domain of f is connected then the index of the operator df (p) is independent of p and by definition is the index of f , cf. Smale [22].
Let η(t) be a geodesic of the L 2 metric (1) in D s ω , emanating from the identity e in the direction v o ∈ T e D s ω . The point η(t * ), t * > 0, is said to be conjugate to η(0) if the linear operator fails to be an isomorphism. If dim ker D exp e (t * v o ) = k , k is called the multiplicity of the conjugate point. A linear operator between Hilbert spaces with empty kernel need not be an isomorphism. Therefore, η(t * ) may be a conjugate point It was shown in [6] that the exponential mapping exp µ,ex e is a non-linear Fredholm map of index zero, in the sense of Smale [23]. In particular, conjugate points in D s µ,ex (M 2 ) are isolated, of finite multiplicity, and the two types of conjugacies coincide.
In this paper we extend the result of [6] and prove We remark that Theorem 1.1 provides a distinction between symplectic diffeomorphisms and Volume preserving diffeomorphisms, when equipped with the L 2 metric. Theorem 1.1 holds for any symplectic manifold of dimension 2n, while Theorem 1.1 fails for the Volume-preserving diffeomorphism group of manifolds of dimension 3 and higher, cf. [6] and [18]. The relationship between Theorem 1.1 and known classifications (e.g. C 0 closure, Gromov's non-squeezing Theorem) of symplectic diffeomorphisms is unclear. The result also provides an interesting contrast to the Hofer metric which is a bi-invariant metric on the group of Symplectomorphisms whose conjugate points do not have such a nice description. See Hofer-Zehnder [8] or Ustilovsky [25].
In a forthcoming paper we show how Theorem 1.1 yields a new characterization of conjugate points in terms of coadjoint orbits. We also show that the notions of Eulerian and Lagrangian stability of the symplectic Euler equations coincide, which completes the results of Preston [17] in that we make no assumptions on the topology of M , nor on the qualitative behaviour of solutions to the symplectic Euler equations.
Denote by T e D s Ham the Lie subalgebra of T e D s ω consisting of globally Hamiltonian vector fields. The algebra T e D s Ham consists of vector fields of the Ham there corresponds a unique element of H s+1 0 . Also, each element of H s+1 0 uniquely determines an element of T e D s Ham . Since T e D s Ham is of finite codimension in T e D s ω it suffices to prove Theorem 1.1 for the exponential mapping restricted to T e D s Ham . Then on T e D s ω the exponential mapping will still be Fredholm.
In section 2 we study the exponential map and it's derivative in terms of solutions to the Jacobi equation, which is simply the linearization of the symplectic Euler equation and the flow equationη = v • η, where v solves the symplectic Euler equation (3). Using a convenient decomposition of the solution operator to the Jacobi equation we show, in section 3, that the derivative of the exponential mapping is the sum of an invertible linear operator and a compact operator and is therefore Fredholm.
2. The Jacobi equation. Let η be the geodesic of (1) starting from the identity in the direction v o ∈ T e D s ω . In order to study Fredholmness of the exponential map it is convenient to express its derivative at tv o in terms of solutions to the Jacobi equation Here, ∇ ω is the right-invariant Levi-Civita connection of (1) given along a geodesic η(t) by for X = u • η, u ∈ T e D s ω and ∇ the Levi-Civita connection on M , cf. [3]. R ω is the right-invariant Riemann curvature tensor of (1), given by The curvature tensore R ω is a bounded multi-linear operator in the H s topology. To see this, let u, v ∈ T e D s ω and define where denotes the Laplace-Beltrami operator. Extending (u, v) H s to T D s ω by right-invariance gives a smooth invariant Riemannian metric on D s ω whose induced topology is equivalent to the underlying topology on D s ω . Let u, v, w ∈ T e D s ω and z ∈ C ∞ (T M ) with L z ω = 0. Since s > dim M 2 + 1 and P is an orthogonal projection onto T e D s ω and ∇ ω is a weak Riemannian connection, we have From the tensorial character of R ω and the fact that the map η → P η is continuously differentiable we similarly obtain The general case follows from right invariance. Consequently, Jacobi fields exist, are unique and global in time along the geodesics in D s ω . If K is the Jacobi field along η with initial conditions (6), then defines a family Φ(t) of bounded linear operators from T e D s ω to T η(t) D s ω . In standard Lie group notation, the group adjoint operator on T e D s ω is Ad η = dR η −1 dL η , where η ∈ D s ω and R η and L η are the right and left translations on D s ω given by the composition with H s diffeomorphisms on the right, respectively the left. Consequently Ad η (X) = Dη · X • η −1 .
(8) is the usual push-forward of vector fields and for the algebra adjoint action See [2] for derivations of formulas (8), (9). The group coadjoint Ad * η : and the Lie algebra coadjoint ad * Proof. Let v G be any vector in T e D s Ham . Using (9) For σ ≥ 0, let T e D σ Ham denote the closure of the space of globally Hamiltonian vector fields in the H σ norm. By the Hodge decomposition this is a closed subspace in the space of all H σ vector fields ( [15]). For σ > dim M 2 + 1 this coincides with the actual tangent space to D σ Ham . However, for smaller σ, D s Ham is not necessarily a smooth manifold.
We have the following decomposition of the solution operator to the Jacobi equation whose proof can be found in [6]. Notice that the decomposition loses one derivative; the equations are only defined on H σ if the initial velocity defining the geodesic is in H σ+1 . However, we will be able to compensate for this by using a smoother geodesic and applying a density argument to obtain the result on H s , s > dim M 2 + 1, s ≥ σ + 1. (5) extends to a continuous linear operator from T e D σ ω to T η(t) D σ ω . In addition, we have the formula where and K v (·) = ad * (·) v and v solves the Euler equation.
Proof. [6] We will also make use of the following convenient formulas for the Hodge projections P and Q.
Lemma 2.2. The projections P and Q, given by orthogonal projection onto the first and second summands of (2), respectively, are given by Proof. For any v ∈ T e D s ω we have ω (v) = dδβ + σ = d −1 δω (v) + σ for some β ∈ H s+2 (T * M ) and σ ∈ H. Hence the orthogonal projection P : T e D s → T e D s ω can be written as where P H denotes the projection onto the finite-dimensional space of harmonic forms which we henceforth neglect. Using the compatible structure J we may further write Q is computed similarly.
3. Proof of Fredholmness. We are now in a position to prove theorem 1.1 by showing that the operator Ω(t) defined in (14) is invertible and the operator Γ(t) defined by (15) (14) is an invertible linear operator on T e D 0 Ham with Here we have used right-invariance of the L 2 metric and the L ∞ denotes the maximum on M of the largest eigenvalue of the symmetric matrix Dη(τ ) T Dη(τ ), which is well defined since η is C 1 . By the Schwartz inequality, So Ω(t) has empty kernel and closed range. Since Ω(t) is also self-adjoint it has empty cokernel and is therefore invertible on T e D 0 Ham . Let O be a coordinate patch on M and H ∈ H σ (O). The Sobolev topology can be defined locally by where α = (α 1 , . . . α n ) is a multi-index with |α| = n j=1 α j and ∂ α = ∂ α1 x1 · · · ∂ αn xn . Lemma 3.2. Suppose M is a compact symplectic manifold of dimension 2n, without boundary, and O a coordinate patch in M . Suppose η ∈ D s ω ,with s ≥ σ + 1 and s > dim M 2 + 1. Then, for any multi-index α with |α| ≤ σ and any w ∈ H σ , we have the estimate for some constant B α .
Proof. Let w ∈ H σ . Then we can write w = v + Jg δα, for some H σ+1 2-form α and v = J∇F ∈ T e D s Ham for some H σ+1 function F . We prove the estimate by induction. Consider We will prove With N j i = ∂η j ∂x i • η −1 , J k l the components of the almost complex structure, g lm the components of the inverse metric, (δα) m the components of the 1-form δα, and , j denoting a derivative in the x j variable,

JAMES BENN
Now (δα) m,j = (δα ,j ) m (since δ = d , with the Hodge star operator) and (which follows from the formula δ(f · γ) = f δγ + i ∇f γ for any function f and any k-form γ). Consequently Since P = J∇ −1 divJ and J 2 = −I, the last term projects to zero. Therefore The L 2 norm of the second term is bounded by the L 2 norm of the first derivatives of α which are, in turn, bounded by the L 2 norm of w. It suffices to bound the first term by the L 2 norm of w.
For any vector field w Q(w) = ω δ −1 dω (w), by Lemma 2.3. Using the Leibniz rule and the fact that dω (v) = 0, the expression (18). Inductively, the estimate (18) implies that for any multi-index α, with no more than σ terms, we will have (17).  (14) is an invertible linear operator on T e D σ Ham with In particular, Ω(t)is invertible on T e D σ Ham . Proof. We have where we have used Lemma 3.1 in the last inequality.
With the definition of Ω(t) we write and use this to show that [∂ α , Ω(t)] v H L 2 is bounded above by the H σ−1 norm of v H . It is enough to show that, for each τ , where dR η P dR η −1 = P η . Now Both Dη −1 and (Dη −1 ) T are matrices of H σ functions. The commutator of a σorder differentiation operator and a multiplication operator is an operator of order σ − 1, by the Leibniz rule. Therefore, the first and third terms above are bounded above by v H H σ−1 . Applying Lemma 3.2 to the second term we obtain (20) and therefore (19).
The estimate (19) shows that Ω(t) has empty kernel (since Ω(t) has empty kernel in L 2 ) and closed range on T e D σ Ham . Choose any v H ∈ T e D σ Ham .
Since Ω(t) is invertible on T e D 0 Ham we can find a v F ∈ T e D 0 Ham such that Ω(t)v F = v H . But now the estimate (19) shows that v F is in H σ as well and Ω(t) is therefore invertible on T e D σ Ham .
We now proceed to prove compactness of the operator Γ(t Ham . We estimate and the Lemma follows.
Lemma 3.5. Let s > dim M 2 + 1, s ≥ σ + 1. For any vector field v F ∈ T e D s Ham the operator K v F defined in Proposition 2.2 is compact on T e D σ Ham . Proof. By Lemma 3.4 we can approximate v F in the H s norm by a sequence of smooth vector fields v F k such that K v F k → K v F in the H σ operator norm. Since a limit of compact operators is compact it suffices to show that K v F is compact when v F is smooth.
By Proposition 2.2 and Lemma 2.1, the operator K v F may be written as since the projection of a gradient field vanishes. Then, for any v H ∈ T e D s Ham , , as a map from H σ vector fields to H σ+1 vector fields, is compact by the Rellich embedding Theorem.
u 0 H s where we have used the estimate (19) of Lemma 3.1. Therefore, with κ(t) = Dη(t) · Γ(t) we obtain the estimate The number C(t) of Lemma 3.1 depends only on the C 1 norm of η, as does Dη L ∞ . So if we chooseṽ F0 close enough to v F0 in the H s norm thenC is close to C, Dη L ∞ is close to Dη L ∞ . Also, Φ(t) −Φ(t) H s is close to zero so that A can be made positive and the estimate (21) is satisfied. Thus Φ(t), and hence d exp e (tv F0 ), has closed range in the H s topology.
This completes the proof of Theorem 1.1.