A Hofer-Type Norm of Hamiltonian Maps on Regular Poisson Manifold

Abstract

We define a Hofer-type norm for the Hamiltonian map on regular Poisson manifold and prove that it is nondegenerate. We show that the -norm and the -norm coincide for the Hamiltonian map on closed regular Poisson manifold and give some sufficient conditions for a Hamiltonian path to be a geodesic. The norm between the Hamiltonian map and the induced Hamiltonian map on the quotient of Poisson manifold by a compact Lie group Hamiltonian action is also compared.

1. Introduction and Main Results

This paper is devoted to establishing an invariant norm for Hamiltonian maps on the Poisson manifold. When is symplectic, a remarkable bi-invariant distance was defined on . This bi-invariant distance was first discovered by Hofer on the group of compactly supported symplectic diffeomorphisms of (where is the standard symplectic form) [1]. Viterbo defined a bi-invariant metric by generating functions [2], Polterovich generalized Hofer’s metric to more symplectic manifold [3], and finally Lalonde and McDuff extended it to the group of compactly supported Hamiltonian diffeomorphisms on any symplectic manifold [4]. This norm plays an important role in studying symplectic topology and has close relationship with symplectic capacity and symplectic rigidity; many mathematicians have great work in this field, but there is few work on the Poisson case; this is because the lack of variational formulation in the Poisson case, it is not easy to prove the nondegenerate. In this paper, we define a Hofer-type norm on a class of Poisson manifolds, that is, regular Poisson manifolds; with the help of Casimir functions and the decomposition of Poisson manifold, we can prove the nondegenerate. Let be a Poisson manifold; that is, there exists a Poisson bracket on the smooth functions . For any it satisfies the following:(1), (2), (3).

Definition 1. A smooth diffeomorphism is called a Poisson diffeomorphism if for all , one has .

Given , the Hamiltonian vector is defined by . Let , be the set of Casimir functions. In this paper, one considers the time-dependent Hamiltonian functions . If the manifold is compact, or the function is compactly supported, then the flow of the Hamiltonian vector globally exists. One denotes by , the set of such Hamiltonian flows and set of time-1 map of such flows, respectively.

For , define If is a Hamiltonian flow with some Hamiltonian function , one defines its length to be one can generalize the bi-invariant metric on to the Poisson case.

Definition 2. Now one can define the energy of , So one can define : for .

Theorem 3. Let be a regular Poisson manifold; the function is a bi-invariant metric; that is, for all it satisfies the following: (1) and if and only if ,(2) and ,(3).

Here a Poisson manifold is called regular if the rank of the Poisson manifold is constant for all point. If one replaces the -norm by the -norm, one also gets a norm on . One proves that they are equal on closed regular Poisson manifold.

Theorem 4. For on a closed regular Poisson manifold, one has

Let be a Poisson manifold and let be a Lie group acting canonically, freely, and properly on via the map . Let be the corresponding optimal momentum map. Then the orbit space is a Poisson manifold with Poisson bracket uniquely characterized by the relation for any and are arbitrary smooth functions.

For a -invariant smooth function on , the Hamiltonian flow of induces a Hamiltonian flow , so one has a well-defined homomorphism where denotes the -invariant Hamiltonian maps. Now one can give a similar result as stated in [5].

Theorem 5. For a -invariant Hamiltonian path with Hamiltonian function , if , one has

Moreover, if the path is length-minimizing, one has the following corollary.

Corollary 6. If the -invariant Hamiltonian path is length-minimizing, then

Organization of This Paper. The organization is as follows. First we will introduce the definition of the distance and give some properties. Next we will show the proofs of Theorems 3 and 4. Then we introduce the Poisson reduction. And last we give the proof of Theorem 5.

2. The Distance on

In this section, we recall the construction of Hofer-norm on and give our definition. For , define .

For , we now define

Proposition 7. defined above is a pseudonorm.

Proof. We just need to show the triangle inequality holds. Let and such that , and for given . Then
From the definition we can see that when , so together with the triangle inequality we get that if and satisfies , then .

Proposition 8. The new pseudonorm is -invariant.

Proof. First, note that is -invariant. If then is a decomposition of . So
So for any , and .
We get that .
Now consider a Hamiltonian function ; the length of the Hamiltonian path generated by can be defined as follows:
This length is well defined; that is, it is independent the choice of the Hamiltonian functions. This is because of our pseudonorm vanishing on .

Definition 9. Now one can define the energy of ,

Proposition 10. The energy function has the following properties: (1) and , (2), (3), (4),
where and is a Poisson diffeomorphism of .

To prove this, we need to investigate the Hamiltonian functions of the Hamiltonian flows. Similar to the symplectic case, for the symplectic case, see page 144 of [6].

Definition 11. If , are smooth functions in and one defines the functions and as follows:

Proposition 12. If , are smooth functions the following formulate hold true: where .

To prove this, we need the following fact.

Lemma 13 (see [7]). If is a Poisson map, and , then

Proof. For any function , we have Since is a Poisson map, we have So we have

Proof of Lemma 13. The third formula is just the transition law of Hamiltonian vector. We know that if is a Poisson map then Now we prove the second formula; we abbreviate the notation and observe that
We need to show that is the flow of ,
By the property of the Poisson diffeomorphism, we get that the second term is . This finishes the proof of the second formula. We can obtain the first formula from the second. From the first two we can get the last one.

We are now ready to prove Proposition 10.

Proof of Proposition 10. From Proposition 12 and the -invariant of our pseudonorm, we get and thus . From we find so the third equality holds.
To prove the last one, we note that , and so we have , and this implies the last one. Now we prove the first one; that is, implies . By definition, . Note that regular Poisson manifold is essentially a union of symplectic manifolds which fit together in a smooth way, so we denote by the symplectic leaf of . The Hamiltonian vector field restricted to each leaf is just the Hamiltonian vector field generated by the restriction of the Hamiltonian function to the leaf. And the Hamiltonian flow keeps the symplectic leaf; that is, , so we can consider the restriction of to each leaf. For each Hamiltonian function generating , we denote by the restriction of on ; for any Casimir function , the restriction of on each leaf is constant. Let be the Hofer norm on the symplectic leaf, Taking the infimum of , we get and hence By the assumption of , we get that , by the definition of Hofer metric, and so .

Now we can define : for . This distance has the following properties.

Theorem . Let be a Poisson manifold; the function is a bi-invariant metric; that is, for all it satisfies the following: (1) and if and only if , (2) and , (3).

Proof. The proof of this theorem is a consequence of Proposition 10.

Example 14. We consider the trivial Poisson manifold ; the Poisson bracket is always zero. In this situation, any function on is Casimir function, the Hamiltonian vector is always zero and the Hamiltonian diffeomorphism is only ; the Hofer-norm of the Hamiltonian map is . When the Poisson manifold is symplectic, that is, there is only one leaf, in this case the Hofer norm is just the one defined by Hofer.

Remark 15. Theorem 3 holds not only for regular manifold, but also for many other manifolds, for example, when the rank of the Poisson manifold is not zero, or the symplectic leaves are always open or always closed.

Next we can get an estimate for the commutators in , denoted by

Claim 16. for .

Proof. From Proposition 10 we get Similarly, so the claim is proved.

Proposition 17. If is open and bounded, satisfies , then for all with supports contained in .

Proof. Define since and , we get that . Consequently The above maps equal . According to Claim 16 and Proposition 10 we get

Proposition 18. For a subset , define the displacement energy of . Then for any open bounded nonempty subset , one has (1), if , then ; (2) for .

Proof. is a consequence of above statements and the monotonicity is by the definition. We just prove the second one. For , if , then ; if , then . According to Proposition 10, we have ; from the above identities, we get the conclusion.

Following [8], we now define a new function for a Poisson map. For a Poisson map on a closed Poisson manifold , define where . Similarly, we can define if we restrict in the closed ball of radius of of centered at .

Definition 19. A map on a closed regular Poisson manifold is bounded if and unbounded otherwise.

Proposition 20. For any , the function is bi-invariant, assumes the value only at , and satisfies the triangle inequality for any Poisson map and .

Proof. The proof is a consequence of the definitions of and the metric.

Proposition 21. Let be a closed regular Poisson manifold; if there exists some unbounded, then the Hofer metric does not extend to a bi-invariant metric on the groups of Poisson maps.

Proof. Assume that we can extend the Hofer metric to the groups of Poisson maps; we still denote the metric by ; then for any . So we have for any , and this is impossible; hence we finish the proof.

Now we consider the geodesic under the above norm in . For the standard symplectic manifold, Hofer proved that, the Hamiltonian flow generated by the time-independent compactly supported function is a geodesic [6]. Later Bialy and Polterovich gave a sufficient and necessary condition for a path to be geodesic [9]; last Lalonde and McDuff extended it to all symplectic manifolds [4]. Now we consider similar questions on regular Poisson manifold.

Theorem 22. Let be a Hamiltonian flow with compactly supported time-independent Hamiltonian function on , and if the maximal and minimal point of lie in the same symplectic leaf, then is a geodesic; that is, for sufficiently small.

Proof. First, by the definition we have To prove the converse, let be the symplectic leaves of the Poisson manifold; we still denote by the restriction of on ; for any compactly supported Hamiltonian function generating , we have
So on each leaf, but on each leaf, according to the results on symplectic manifold mentioned above, the flow is a geodesic; we thus have for sufficiently small. By the assumption, we have for some leaf . Hence Since is compactly supported, we have the -norm of is bounded above by some number , and moreover, so by Theorem 1.2 of [3], we can choose a constant such that when and this finishes the proof.

For the time-dependent case, we have a similar result. First we recall the definition of quasiautonomous function.

Definition 23. A function on is called quasiautonomous if there exist two points such that , for all .

Theorem 24. Let be a Hamiltonian flow with compactly supported Hamiltonian function on ; if is quasiautonomous on each symplectic leaf, and the fixed maximum and fixed minimum of lie in the same symplectic leaf, then is a geodesic; that is, for sufficiently small.

Proof. We first adopt the transformation in [10] to simplify the problem; for interval , define and for , . Then the Hamiltonian flow of satisfies This implies that ; by the assumption, is quasiautonomous on each symplectic leaf and so is ; by Theorem 1.2 of [11, 12], we know that is a minimum geodesic on each leaf provided that sufficiently small; that is, Because the fixed maximum and fixed minimum of lie in the same symplectic leaf, we may assume that this leaf is . We have So,
By Theorem 3, we have .

Remark 25. If the Poisson manifold is symplectic, then Theorems 22–24 reduce to the results in [4, 6, 9]. If we make more assumptions on the Hamiltonian functions, we can get similar results about the minimizing geodesics.

Theorem 26. Let be a sequence of smooth Hamiltonians on a closed Poisson manifold ; suppose that (1) in the -topology,(2) in the -topology, .
If all are minimizing, then is also minimizing.

Proof. We employ the method of Oh in [13] to prove it; for reader's convenience, we write it here. Suppose that is not minimizing, then we choose a function , such that ; choose such that then when is sufficiently large. Define By simply computations as we know that , and in the -topology, so for any Casmir function , we have , and thus This is a contraction and we finish the proof.

Theorem 27. Assume that and , and is a homeomorphism of . If (1),(2), locally uniformly,
then .

Proof. We assume that , and by the assumption we have and . We restrict them to each leaf and adapt the same notations and arguments in Proposition 10; we have and . By Theorem 6 page 169 in [6], we know that ; this holds on each leaf, so .

Corollary 28. If and is a homeomorphism of satisfying (1) uniformly, (2), locally uniformly
then .

If one replaces the -norm by the -norm, one also gets a pseudonorm on .

For a Hamiltonian function , define the pseudolength of the Hamiltonian path generated by as follows: Similarly, one can define the energy and the pseudometric.

Definition 29. The energy of , Define : for . This is also a bi-invariant metric. One denotes by , the induced -norm and the -norm, respectively.
Recall that in the symplectic case, Polterovich proved that the -norm and the -norm coincide on closed symplectic manifolds [14]. We now give a similar result in the Poisson case.

Theorem . For on a closed Poisson manifold, one has

We first show the following results which will be useful in the proof.

Proposition 30. is closed in the -topology.

Proof. We just show that this is true on each symplectic leaf, but in the symplectic case, the Casimir functions are constants; this finishes the proof.

Proposition 31. Let be a flow generated by a Hamiltonian function on a closed Poisson manifold. Then there exists an arbitrary small loop such that the Hamiltonian function of the flow satisfies for every .

Proof. First, by Proposition 12 we know that the Hamiltonian function of can be given by . We now just show that for all t. Take a function such that for every . Then define as the time- map of the Hamiltonian flow generated by . Let be the Hamiltonian function of the loop , then we need the following proposition of Banyaga.

Proposition 32 (Banyaga, cf. Proposition 3.1.5 [15]). Let be a 2-family smooth parameters of diffeomorphisms on a smooth manifold such that . Let be the families of vectors on defined by then

By the above proposition, we now compute . By definition So up to a Casimir function.

Now we define

Here every is constructed with the help of as above; we know that the partial derivative of the Hamiltonian with respect to at equals .

Fixed a regular point , let be the symplectic leaf though , consider the linear space . Choose smooth closed curves (where ) satisfying the following conditions:(1) for all ;(2)the vectors are linearly independent for every . The existence of such system of curves is shown in [14]; for example, choose a basis in and take the curves of the form and .

Now choose such that(1) for every ;(2) for every ;(3).

Take the corresponding -parameter variation of the constant loop as above. Consider the map defined by It follows that is a submersion in some neighbourhood of the circle . Indeed from our construction we have

But these vectors generates the whole . Denote by the restriction of to . Since is a submersion, the set is a one-dimensional submanifold of , so there exist arbitrary small values of the parameter such that for all . This completes the argument.

Proof of Theorem 4. For , clearly we have . Now we prove the converse. Fix a positive number , choose a path such that , with Hamiltonian functions and . By Proposition 31 we can assume that for all since the manifold is good manifold. Here this pseudonorm is defined in Proposition 7. Define as the inverse of where . Note that the Hamiltonian of can be generated by , where denotes the derivative with respect to . Note that we get that . Approximating in the -topology by a smooth one, we get a Hamiltonian denoted by ; we have . Since is arbitrary, we conclude that . This completes the proof.

Corollary 33. For , one has

Proof. Let be another Hamiltonian flow with Hamiltonian and satisfy ; then there exists a Hamiltonian function such that and . On the other hand, for each loop , the time one map of the flow is also ; by Proposition 12 its Hamiltonian can be Note that generates loop , so

3. Poisson Reduction

In this section, we briefly introduce the Poisson reduction. Let be a Lie group acting canonically on ; if the action is free and proper, we know that the orbit space is a smooth manifold and the canonical projection is a smooth surjective submersion. Let be the corresponding optimal momentum map. The orbit space is a Poisson manifold with Poisson bracket uniquely characterized by the relation for any and are arbitrary smooth functions.

The Poisson structure induced by the bracket on is the only one for which the projection is a Poisson map. Let be a -invariant Hamiltonian flow of commutes with the -action, so it induces a flow on characterized by The flow is Hamiltonian on for the reduced Hamiltonian function defined by The vector fields and are -related.

So we have a well-defined homomorphism where denotes the -invariant Hamiltonian maps.

More details can be found in [16].

4. Proof of Theorem 5

Now we can give the proof of Theorem 5.

Theorem . For a -invariant Hamiltonian path with Hamiltonian function , if , one has

Proof. From the above discussion, we know that For any -invariant Hamiltonian path , it induces a Hamiltonian path on .
Let be the Hamiltonian of the Hamiltonian path , and the induced path . We have
By the definition of the norm, we have and note that According to the above discussions, we have so
Moreover, if the path is length-minimizing, that is, then we have Corollary 6.

Remark 34. If the Poisson manifold is symplectic, then the pseudonorm is the Hofer norm. Give a Hamiltonian -action; then we can get the results in the symplectic case as in [5].

Conflict of Interests

The authors declare that they have no conflicts of interest regarding this work.

Acknowledgments

The authors would like to express their deep gratitude to Professor Yiming Long for many valuable discussions. The research was supported by TianYuan Program of National Natural Science Foundation of China (11226158), Natural Science Foundation of Henan (2011B110011), and Doctor Fund of Henan University of Technology.