Gromov’s alternative, contact shape, and
-rigidity of contact diffeomorphisms
We prove that the group of contact diffeomorphisms is closed in the group of all diffeomorphisms in the -topology. By Gromov’s alternative, it suffices to exhibit a diffeomorphism that can not be approximated uniformly by contact diffeomorphisms. Our construction uses Eliashberg’s contact shape.
2010 Mathematics Subject Classification:53D10, 53D35, 57R17
1. Introduction and main result
Let be a connected symplectic manifold, and be the corresponding canonical volume form. We assume throughout this paper that all diffeomorphisms and vector fields are compactly supported in the interior of a given manifold.
A symplectic diffeomorphism obviously preserves the volume form , and thus so does a diffeomorphism that is the uniform limit of symplectic diffeomorphisms. In the early 1970s, Gromov discovered a fundamental hard versus soft alternative [Gro86]: either the group of symplectic diffeomorphisms is -closed in the group of all diffeomorphisms (hardness or rigidity), or its -closure is (a subgroup of finite codimension in) the group of volume preserving diffeomorphisms (softness or flexibility). See section 2 for a more precise statement.
Gromov’s alternative is a fundamental question that concerns the very existence of symplectic topology [MS98, DT90]. That rigidity holds was proved by Eliashberg in the late 1970s [Eli82, Eli87]. One of the most geometric expressions of this -rigidity is Gromov’s non-squeezing theorem. Denote by a closed ball of radius in with its standard symplectic form , and by a cylinder of radius so that the splitting is symplectic.
Theorem 1 (Gromov’s non-squeezing theorem [Gro85]).
If there exists a symplectic embedding of into , then .
It follows at once that is -closed in . Using Darboux’s theorem, one can easily deduce that rigidity holds for general symplectic manifolds.
The group is -closed in the group of all diffeomorphisms.
See section 2 for details. It was observed later that symplectic capacities give rise to a -characterization of symplectic diffeomorphisms, and in particular to proofs of Theorem 2 that do not rely on Gromov’s alternative [MS98, HZ94]. Note that the existence of symplectic capacities is equivalent to Theorem 1.
Let be a connected contact manifold of dimension , that is, is a completely non-integrable codimension one tangent distribution. In this paper, all contact structures are assumed to be coorientable, i.e. there exists a -form with and is a volume form on . Moreover, we always fix a coorientation of , so that the contact form is determined up to multiplication by a positive function, and unless explicit mention is made to the contrary, a contact diffeomorphism is assumed to preserve the coorientation of .
In the contact case, the analog of Gromov’s alternative is the following: the group of contact diffeomorphisms is either -closed or -dense in . See section 2 for a precise statement. A popular folklore theorem among symplectic and contact topologists is that an analog of Theorem 2 holds in the contact case. However, up to this point no proof has been published. The purpose of this paper is to remedy this situation by proving the following main theorem.
Theorem 3 (-rigidity of contact diffeomorphisms).
The contact diffeomorphism group is -closed in the group of all diffeomorphisms. That is, if is a sequence of contact diffeomorphisms of that converges uniformly to a diffeomorphism , then preserves the contact structure .
Similar to the case of symplectic diffeomorphisms, to prove Theorem 3 we single out an invariant of contact diffeomorphisms that is (at least in the special case that is considered below) continuous with respect to the -topology. In section 4 we will use the contact shape defined in [Eli91] to construct a diffeomorphism that cannot be approximated uniformly by contact diffeomorphisms. The construction is local and therefore applies to any contact manifold. We recall Eliashberg’s shape invariant in section 3. We suspect that there are alternate proofs of Theorem 3 based on other contact invariants. It would also be interesting to discover a -characterization of contact diffeomorphisms (cf. [MS98]) similar to that of (anti-) symplectic diffeomorphisms, and then to find a proof of contact rigidity that does not use Gromov’s alternative.
To make some final remarks, let and be as in the rigidity theorem. Then and for smooth functions and on . The hypothesis of Theorem 3 does not guarantee that the sequence converges to even pointwise, see [MS11, MS12]. The following theorem is a special case of a contact rigidity theorem that is proved in the context of topological contact dynamics in [MS11].
Theorem 4 (-rigidity of contact diffeomorphisms and conformal factors [Ms11]).
Let be a contact manifold with a contact form , and be a sequence of contact diffeomorphisms of that converges uniformly to a diffeomorphism , such that , and converges to a (a priori not necessarily smooth) function uniformly. Then is a contact diffeomorphism, the function is smooth, and . In particular, if and are two contact isotopies that are generated by smooth functions and on , respectively, then for all times if and only if for all .
We would like to thank Yasha Eliashberg for many helpful discussions regarding the proof of -rigidity of contact diffeomorphisms, in particular during a recent conference at ETH Zürich in June 2013.
2. Gromov’s alternative
Recall that the group of Hamiltonian diffeomorphisms is comprised of those diffeomorphisms that are time-one maps of (time-dependent) vector fields such that the -forms are exact for all times .
Theorem 5 (Gromov’s maximality theorem [Gro86, Section 3.4.4 H]).
Let be a connected symplectic manifold. Suppose that a (not necessarily closed) subgroup contains all Hamiltonian diffeomorphisms, and that contains an element that is neither symplectic nor anti-symplectic, i.e. . Then contains the subgroup of volume preserving diffeomorphisms with zero flux.
We do not recall the flux map in detail, but note that it is a homomorphism from the connected component of the identity of into a quotient of the group by a discrete subgroup. Its kernel is comprised of those diffeomorphisms that are generated by (time-dependent) exact vector fields , i.e. such that the -form is exact for all times . In particular, the time-one map of a divergence-free vector field that is supported in a coordinate chart has vanishing flux. The flux of a Hamiltonian diffeomorphism is also equal to zero. See [Ban97] for details.
The easier part of the proof uses linear algebra to prove that there are enough diffeomorphisms so that every exact vector field can be decomposed uniquely into a sum , where the vector field is Hamiltonian with respect to the symplectic form . Then the linearization of the map , , is surjective near the identity, and also has a right inverse. The same techniques used to prove Gromov’s difficult implicit function theorem then show that the image of contains an open neighborhood of the identity in the subgroup of all diffeomorphisms that are generated by (time-dependent) exact vector fields. But this open neighborhood generates the entire subgroup of diffeomorphisms with zero flux, hence the proof. The assumption that all vector fields are compactly supported in the interior guarantees that their flows are well-defined and exist for all times.
In order to apply Gromov’s maximality theorem in the proof of Theorem 2, we need the following lemma.
Let be a connected symplectic manifold. Then the -closure of the group of symplectic diffeomorphisms in the group of all diffeomorphisms of does not contain any anti-symplectic diffeomorphisms.
Suppose that is not closed, and for a diffeomorphism . Since the sign on the right-hand side is independent of , and every diffeomorphism is equal to the identity outside a compact subset of the interior of , cannot be anti-symplectic.
Next suppose that is closed. Since the group of homeomorphisms of is locally contractible with respect to the -topology, any two homeomorphisms that are sufficiently -close induce the same action on the real cohomology groups of . In particular, the closure of in fixes the non-zero cohomology class , and thus it does not contain any anti-symplectic diffeomorphisms.
For an alternate proof for closed symplectic manifolds, recall that the -closure of is contained in the group of volume preserving diffeomorphisms. But if , then , and thus if is odd, the group does not contain any anti-symplectic diffeomorphisms. Then one proceeds to first proving Theorem 2 for odd, and proves Theorem 2 for even afterward.
If is even, one applies the above argument to a product manifold , where is a closed surface with an area form . Assume a diffeomorphisms is contained in the -closure of , and let be a sequence of symplectic diffeomorphisms that converges uniformly to . Since is odd, is -closed in by what we have already proved, and in particular the -limit of the sequence lies in . But then must preserve . ∎
For , one obtains the common formulation of Gromov’s alternative.
The group of symplectic diffeomorphisms is either -closed in the group of all diffeomorphisms of , or its -closure is the full group of volume preserving diffeomorphisms.
If is not -closed, then by Gromov’s maximality theorem and Lemma 6, it contains the connected component of the identity in . But the -closure of the latter generates the entire group. Indeed, let denote the map , and let be a compactly supported volume preserving diffeomorphism of . Then for sufficiently small, the time- map of the volume preserving isotopy is supported in a small neighborhood of the origin, and thus is -close to the identity. ∎
Proof of Theorem 2.
Let be the domain of a Darboux chart, and identify with an open subset of with its standard symplectic structure. For sufficiently small, there exists a closed ball , and a volume preserving diffeomorphism of that is the identity near the boundary, so that the image of is contained in . This diffeomorphism can be chosen to be generated by a divergence-free vector field with support in , and in particular, it can be viewed as an element of .
Suppose there exists a sequence of symplectic diffeomorphisms of that converges uniformly to . Their restrictions to are symplectic embeddings into for sufficiently large. But that contradicts Gromov’s non-squeezing theorem. Then by Gromov’s maximality theorem, the -closure of cannot contain any diffeomorphisms that are neither symplectic nor anti-symplectic. Finally by Lemma 6, is -closed inside the group . ∎
Theorem 8 (Gromov’s maximality theorem [Gro86, Section 3.4.4 H]).
Let be a connected contact manifold. Suppose that a (not necessarily closed) subgroup contains all contact diffeomorphisms that are isotopic to the identity through an isotopy of contact diffeomorphisms, and that contains an element that does not preserve the (un-oriented) contact distribution , i.e. . Then contains the connected component of the identity of the group .
Again the first step in the proof uses linear algebra to show that there are enough diffeomorphisms so that every vector field can be decomposed uniquely into a sum , where the vector field is now contact with respect to the contact structure . The linearization of the analogous map , , is again surjective near the identity, and has a right inverse. The techniques of Gromov’s implicit function theorem imply that the image of contains an open neighborhood of the identity in the subgroup of all diffeomorphisms generated by (time-dependent) vector fields, and this neighborhood again generates the entire connected component of the identity in the group of all diffeomorphisms of .
Let be a connected contact manifold. The -closure of the group of contact diffeomorphisms in the group does not contain any diffeomorphisms that reverse the coorientation of .
If is not closed, the argument is the same as in the symplectic case.
If is closed, one may use the fact that every element of preserves the orientation induced by the volume form (which is independent of the choice of contact form with ), and that this property is preserved under -limits. If is odd, it follows at once that the -closure of does not contain any diffeomorphisms that reverse the coorientation of . Then one proceeds to proving the main theorem for odd, and again apply it in the proof for even .
If is even, one needs in addition a theorem of Bourgeois [Bou02] that for a closed contact manifold , the product also admits a contact structure with the property that for each point , the submanifold is contact and moreover contact diffeomorphic to . Then one concludes the main theorem for even by the same argument as in the symplectic case. ∎
The group of contact diffeomorphisms is either -closed or -dense in the group of all diffeomorphisms of .
The diffeomorphism preserves the standard contact structure , and thus the proof is similar to the symplectic case. ∎
In the case of with its standard contact structure, it is possible to use non-squeezing phenomena [EKP06] similar to Theorem 1 to decide Gromov’s alternative in favor of rigidity. However, the domain of a Darboux chart in does not contain a subset of the form for an open set , so that this proof does not generalize to arbitrary contact manifolds. If is a three-manifold, then there are non-squeezing phenomena ([Gro86, Section 3.4.4 F] or [EKP06]) that give rise to an alternate proof of -rigidity of contact diffeomorphisms.
3. The contact shape
In this section we review the contact shape defined and studied in [Eli91]. We only present the elements of this invariant that are necessary to prove contact rigidity in the next section, and refer the reader to the original paper for a more general discussion.
Let be an -dimensional torus, and its cotangent bundle with the canonical symplectic structure , where , and where and denote coordinates on the base and the fiber , respectively. An embedding into an open subset is called Lagrangian if , and the cohomology class in is called its -period. Choose a homomorphism . The -shape of is the subset of that consists of all points such that there exists a Lagrangian embedding with and .
Let be a connected contact manifold, and denote by the symplectization of with the symplectic structure , where is the projection to the first factor, and is the coordinate on the factor . (The symplectization can also be defined as the manifold with coordinate on the second factor and the change of coordinates .) Up to symplectic diffeomorphisms, the symplectization depends only on the contact structure , and not on the particular choice of contact form . A contact embedding with induces an (-equivariant) symplectic embedding given by .
Let denote the unit cotangent bundle of with its canonical contact structure , where is the restriction of the canonical -form on . The symplectization of is diffeomorphic to minus the zero section with its standard symplectic structure . Denote by an open subset of . Since is canonically isomorphic to , a given homomorphism can be identified with a homomorphism . Thus the symplectic shape of the symplectization is a contact invariant of the contact manifold . It is easy to see from the definition that is a cone without the vertex in . Thus it is convenient to projectivize (in the oriented sense of identifying vectors that differ by a positive scalar factor) the invariant, and define the contact (-)shape of by
Proposition 11 ([Eli91, Proposition 3.2.2]).
Let and be two open subsets of the unit cotangent bundle with its canonical contact structure, and be a contact embedding. Then . In particular, if is a contact diffeomorphism, then .
The above decomposition of restricts to the decomposition . Choose cohomology classes as a basis of to identify it with the fiber of the fibration . This gives rise to an identification of the (oriented) projectivized group with the fiber of the fibration .
Proposition 12 ([Eli91, Proposition 3.4.1]).
Let be a connected open subset, and . For , denote by the canonical embedding. Then .
4. -rigidity of contact diffeomorphisms
In this section we construct a diffeomorphism that is an element of the connected component of the identity in the group of diffeomorphisms of , but on the other hand is not contained in the -closure of the group of contact diffeomorphisms. Together with Gromov’s maximality theorem and Lemma 9, this yields a proof of the main theorem.
Recall that for closed manifolds, the -topology is given by the compact-open topology. It is also a metric topology induced by uniform convergence with respect to some auxiliary Riemannian metric. (This is not a complete metric, but the -topology is also induced by a complete metric.) For open manifolds, the -topology is defined as a direct limit of the -topologies on compact subsets of the interior.
As in the previous section, denote by the unit cotangent bundle of an -dimensional torus with its standard contact structure induced by the contact form , where denote coordinates on the sphere, and are coordinates on the torus.
Let be a contact manifold, and be a point. Then there exists a neighborhood of in and a contact embedding of with its standard contact structure into .
Since contact diffeomorphisms act transitively on the underlying manifold, it suffices to prove the lemma for a point with for all . We will show that a neighborhood of is contact diffeomorphic to an open subset of with the contact structure induced by the contact form , where , and and are coordinates on . This form is diffeomorphic to the standard contact form . We moreover prove that a neighborhood of in is contact diffeomorphic to an open subset of . Combining the two maps gives rise to the desired embedding.
By Darboux’s theorem, contains an embedded transverse knot . By the contact neighborhood theorem, a sufficiently small neighborhood of this knot is contact diffeomorphic to an open neighborhood of with its standard contact structure. That proves the second claim.
Let be the open subset of on which for all , and define a function by . The map given by
is a contact diffeomorphism onto the subset of on which all spherical coordinates are positive. Since is non-empty, the proof is complete. ∎
Proof of Theorem 3.
By the lemma, the contact manifold contains an open subset that is diffeomorphic to with its standard contact structure. Let be an open ball with center and radius with respect to the standard metric on , and be the time-one map of a smooth vector field that is compactly supported in , and maps the ball to the ball . The diffeomorphism is compactly supported in , and thus extends to a diffeomorphism of that is isotopic to the identity. We will use Eliashberg’s shape invariant to show that can not be approximated uniformly by contact diffeomorphisms. Then by Gromov’s maximality theorem combined with Lemma 9, the -closure of does not contain any diffeomorphisms that do not preserve the (oriented) contact structure .
Suppose is a sequence of contact diffeomorphisms that converges uniformly to . For sufficiently large, the restriction of to is a contact embedding into , and thus by Proposition 11, the shape invariant satisfies the relation . For sufficiently large, the map is homotopic to the restriction of to , and the latter fixes the submanifold . Thus the induced map equals . In particular,
by Proposition 12. On the other hand, , and we arrive at a contradiction. That means the sequence cannot exist, hence the proof. ∎
As a final remark, note that Theorem 3 implies that the set of diffeomorphisms that preserve but reverse its coorientation is also -closed in the group of all diffeomorphisms. Indeed, if this set is non-empty, then it is equal to for some diffeomorphism that reverses the coorientation of .
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