Noncontractible loops of symplectic embeddings between convex toric domains
Given two –dimensional ellipsoids whose symplectic sizes satisfy a specified inequality, we prove that a certain loop of symplectic embeddings between the two ellipsoids is noncontractible. The statement about symplectic ellipsoids is a particular case of a more general result. Given two convex toric domains whose first and second ECH capacities satisfy a specified inequality, we prove that a certain loop of symplectic embeddings between the two convex toric domains is noncontractible. We show how the constructed loops become contractible if the target domain becomes large enough. The proof involves studying certain moduli spaces of holomorphic cylinders in families of symplectic cobordisms arising from families of symplectic embeddings.
1.1 Previous results and a new result about ellipsoids
Questions about symplectic embeddings of one symplectic manifold into another have always been one of the main study directions in symplectic geometry. The pioneering work of Gromov in [gromov1985pseudo] introduced new methods that made it possible to answer many open questions about symplectic embeddings that had been until then unanswered. The survey by Schlenk, [schlenk2018symplectic], presents in detail the type of results one can prove about symplectic embeddings together with the tools used to prove such results.
Most of the questions that have been answered (in the positive or the negative) concern the existence of symplectic embeddings of one symplectic manifold into another. For example, see [mcduff1991blow], [mcduff2009symplectic], [mcduff1994symplectic], and [mcduff2012embedding] for symplectic embeddings involving –dimensional ellipsoids, see [choi2014symplectic], [cristofaro2014symplectic], [cristofaro2017symplectic], and [hutchings2016beyond] for symplectic embeddings involving more general –dimensional symplectic manifolds, and also see [hind2013ellipsoid], [guth2008symplectic], and [gutt2017symplectic] for results in higher dimensions.
Another direction where significant progress has been made is the study of the connectivity of certain spaces of symplectic embeddings. In [mcduff2009symplectic], McDuff shows the connectivity of spaces of symplectic embeddings between –dimensional ellipsoids, while in [cristofaro2014symplectic], Cristofaro–Gardiner extends this result to symplectic embeddings from concave toric domains to convex toric domains, both of which are subdomains of whose definition we recall below in §1.2. In [hind2013symplectic], Hind proves the non-triviality of for spaces of symplectic embeddings involving certain –dimensional polydisks, extending a result that was initially proved in [floer1994applications]. In [gutt2017symplectically], the authors prove that certain spaces of symplectic embeddings involving more general –dimensional symplectic manifolds are disconnected, while in [mcduff1993camel], the authors study the connectivity of symplectic embeddings into generalized “camel” spaces in higher dimensions, extending results in [eliashberg1991convex].
Following yet another direction, in this paper we study the fundamental group of certain spaces of symplectic embeddings in dimensions. Let us first clarify the notation we will be using. For real numbers and with , the set
together with the restriction of the standard symplectic form from is called a closed symplectic ellipsoid, or more simply an ellipsoid. Moreover, we define the symplectic ball . Also, if and are symplectic manifolds, let denote the space of symplectic embeddings of into .
Here are a few results about the fundamental group of spaces of symplectic embeddings that motivated our work. The first result in this direction is an immediate consequence of the methods that Gromov introduced in [gromov1985pseudo] in order to prove the nonsqueezing theorem.
Theorem 1.1 ([eliashberg1991convex]).
Let be an embedded unknotted –sphere in . Write and let be the evaluation map . Then the induced homomorphism is surjective for and trivial otherwise.
Another situation where the fundamental group of a space of symplectic embeddings can be computed is the following.
Theorem 1.2 ([hind2013symplectormophism]).
If the space deformation retracts to .
A more recent result that is closer in spirit to the results of this paper can be found in [burkard2016first], where the author constructs a loop in and shows that if the positive real numbers , , and satisfy , , and , then the constructed loop is noncontractible in . Moreover, the loop becomes contractible if .
By contrast to [burkard2016first], we study symplectic embeddings whose domain is connected. More specifically, this paper is concerned with the study of restrictions of the loop of symplectic linear maps defined in (1) below to certain domains in .
Let denote the loop of symplectic linear maps
The loop is a concatenation of the counterclockwise rotation in the –plane followed by the clockwise rotation in the –plane. The loop is contractible in , but it restricts to give some noncontractible loops of symplectic embeddings. For example:
Assume that and . Then, for defined as in (1), the loop of symplectic embeddings is noncontractible in .
The method of proof we present in §4 does not answer whether the loop is contractible or not under the following assumption.
Open question 1.5.
Assume . Is the loop contractible in ?
1.2 Main theorem
We begin by recalling an important example of –dimensional symplectic manifolds with boundary, in order to prepare for the statement of the main theorem. Given a domain , we define the toric domain
which, together with the restriction of the standard symplectic form on , is a symplectic manifold with boundary. For example, if is the triangle with vertices , and , then is the ellipsoid defined above, while if is the rectangle with vertices , , , and , then is the polydisk . Note that we allow domains that have non-smooth boundary. The toric domains we work with in this paper have the following particular property.
A convex toric domain is a toric domain defined by
such that its defining function is nonincreasing and concave.
Even though we will not work with this type of domains in this paper, let us also recall that a concave toric domain is a toric domain defined also by (3) such that its defining function is nonincreasing, convex, and . For example, ellipsoids are the only toric domains that are both convex and concave, and polydisks are convex toric domains. We next explain how to compute the embedded contact homology (ECH) capacities of convex toric domains in order to state the main result of this paper.
Given a –dimensional symplectic manifold with contact boundary , its ECH capacities are a sequence of real numbers
constructed using a filtration by action of the ECH chain complex . The ECH capacities obstruct symplectic embeddings, meaning that if there exists a symplectic embedding then for all . In particular, for the first and second ECH capacities of a convex toric domain, we can use the following explicit formulas, see [hutchings2016beyond, Proposition 5.6] for details.
For a convex toric domain with nice defining function ,
where is the unique point where .
For the definition of a nice defining function, see §2.4. Every defining function can be perturbed to be nice. Having introduced all the ingredients, we are ready to state the main result of this paper.
Let and be convex toric domains with defining functions and , respectively. Assume that , , and . Then, for defined as in (1), the loop of symplectic embeddings is noncontractible in .
If the target is large enough, the loop becomes contractible, see Figure 4.
Assume there exists such that . Then the loop is contractible in .
Since the loop is contractible in , there exists a homotopy of unitary maps contracting it, where denotes the closed unit disk. For each , the operator norm of is , and hence . So the –parameter family of restrictions is contained in and provides a homotopy from to the constant loop. ∎
1.3 Strategy of proof and the organization of the paper
We use the following strategy to prove Theorem 1.8. For each symplectic embedding , we add to the compact symplectic cobordism (, a positive cylindrical end at and a negative cylindrical end at , in order to construct the completed symplectic cobordism . After choosing an almost complex structure that is compatible with the cobordism structure on , we define the moduli space which consists of –holomorphic cylinders in that have a positive end at the shortest Reeb orbit on and a negative end at the shortest Reeb orbit on .
Using automatic transversality together with a compactness argument which works under the hypothesis of Theorem 1.8, we show that for each and for each compatible almost complex structure , the moduli space is a finite set. We directly construct an almost complex structure and a –holomorphic cylinder with the right asymptotics, to show that is nonempty for the restriction of the inclusion map and the particular choice of . We describe the cylinders near their asymptotic ends to prove that, whenever nonempty, contains a unique –holomorphic cylinder.
We complete the proof using an argument by contradiction. We assume the loop is contractible by the homotopy , for each . We choose a –parameter family of almost complex structures so that is compatible with the cobordism structure on and for all . We define the universal moduli space and, using parametric transversality for generic families of almost complex structures, we show that, for a generic choice of as above, the moduli space is a –dimensional manifold. Assuming the bounds in the hypothesis of Theorem 1.8, we conclude using SFT compactness and the description of each that is homeomorphic to the closed disk .
For the final details, we fix a parametrization of the shortest Reeb orbits on together with a point on the same Reeb orbit. For each , we trace, on the unique cylinder , the vertical ray that is asymptotic to at and record the point where it lands at on the shortest Reeb orbit on . We then study the composition of maps
and show that this circle map has degree . This provides the contradiction we are looking for, since we previously showed that is homeomorphic to the closed disk .
The paper is divided in sections as follows. In §2, we classify the embedded Reeb orbits on the boundary of a convex toric domain. We make use of this classification, together with an automatic transversality argument, to prove the compactness of the moduli space in §3. We also use the classification in §2 to show the compactness of the moduli space in §4.3. Finally, §4.1 contains the argument for the existence of –holomorphic cylinders with the right asymptotics, §4.2 contains the argument for the uniqueness of –holomorphic cylinders in , and §4.3 presents the details behind the construction of the circle map above, in order to complete the proof.
Acknowledgements. I would like to thank my advisor, Michael Hutchings, for all the help and ideas he shared with me. I would also like to thank Chris Wendl for clarifying some of my mathematical confusions during my visit at Humboldt–Universität zu Berlin. Finally, I would like to thank my friends, Julian Chaidez and Chris Gerig, for the many helpful conversations we had.
2 Reeb dynamics and the ECH index
2.1 Geometric setup
Let be a closed –dimensional contact manifold with contact form , i.e. . The Reeb vector field corresponding to is uniquely defined as the vector field satisfying and . A Reeb orbit is a map for some , modulo translations of the domain, such that . The action of a Reeb orbit is defined by and is also equal to the period of .
For a fixed Reeb orbit , the linearization of the Reeb flow of induces a symplectic linear map , called the linearized return map. A Reeb orbit is called nondegenerate if its linearized return map does not have as an eigenvalue. We call elliptic if the eigenvalues of are complex conjugate on the unit circle, positive hyperbolic if the eigenvalues of are real and positive, and negative hyperbolic if the eigenvalues of are real and negative. A contact form is called nondegenerate if all its Reeb orbits are nondegenerate.
2.2 Reeb dynamics on
In this section we compute the Reeb dynamics on the boundary of convex toric domain. Recall that a convex toric domain is defined by (2), with defining set given by (3). Similarly to the computations in [hutchings2014lecture, §4.3], we choose scaled polar coordinates on to obtain
The radial vector field
is a Liouville vector field for defined on all . The boundary of the toric domain is transverse to and so
restricts to a contact form on . The Reeb vector field corresponding to has the following expression. In the two coordinate planes, is given by
While if for some with , , then
The embedded Reeb orbits or are classified as follows:
The circle is an embedded elliptic Reeb orbit with action .
The circle is an embedded elliptic Reeb orbit with action .
For each with , the torus
is foliated by a Morse-Bott circle of Reeb orbits. If with relatively prime positive integers, then we call this torus and we compute that each orbit in this family has action .
The existence of Morse-Bott circles of Reeb orbits implies that the contact form is degenerate. We need to perturb it in order to make it nondegenerate since the nondegeneracy allows the study of –holomorphic curves with cylindrical ends asymptotic to Reeb orbits.
For each , we can perturb to a nondegenerate , where , so that each Morse-Bott family that has action becomes two embedded Reeb orbits of approximately the same action, more specifically an elliptic orbit and a hyperbolic orbit . Moreover, no Reeb orbits of action are created and the Reeb orbits and are unaffected.
Such a perturbation of the contact form is equivalent to a perturbation of the hypersurface on which the restriction of becomes nondegenerate.
2.3 ECH index
Embedded contact homology (ECH) is an invariant for –dimensional contact manifolds due to Hutchings. See [hutchings2014lecture] for a detailed account of history, motivation, construction, and applications of ECH. We give a brief overview of the definition of ECH following the notation from [hutchings2009embedded].
Let be a contact –dimensional manifold with nondegenerate contact form . Given a convex toric domain , the boundary together with a perturbation of , as in Remark 2.1, is such a contact manifold.
An orbit set is a finite set of pairs , where are distinct embedded Reeb orbits and are positive integers. We will also use the multiplicative notation for an orbit set . Denote by the sum and define the action of by . If and are two orbit sets with , then define to be the set of relative homology classes of –chains such that . Note that is an affine space over .
Given a , define the ECH index of by the formula
where is a choice of symplectic trivializations of over the Reeb orbits and , denotes the relative first Chern class (see [hutchings2009embedded, §2.5]), denotes the relative self-intersection number (see [hutchings2009embedded, §2.7]), and
where is the Conley–Zehnder index with respect to of the orbit (see [hutchings2009embedded, §2.3]).
The ECH index does not depend on the choice of symplectic trivialization. The definition of the ECH index can be extended to symplectic cobordisms by generalizing the definitions of the relative first Chern class and of the self intersection number (see [hutchings2009embedded, §4.2]).
If and , then . In the particular case of starshaped hypersurfaces in , this implies there is an absolute grading on orbit sets as follows. Since , for every pair of orbit sets and there is an unique class . Define for the empty orbit set and set
where is the unique element of . Also, let and .
2.4 Absolute grading on
Following the details in [hutchings2016beyond, §5], we recall the classification of the orbit sets on the boundary of a convex toric domain that have ECH index .
Similarly to [hutchings2016beyond, Lemma 5.4], we first perform a perturbation of the geometry of (see Figure 5). This means we can assume, without loss of generality, that the function defining is nice, meaning that satisfies the following properties:
is irrational and is approximately ,
is irrational and is very large, close to ,
except for in small connected neighborhoods of and .
Lemma 2.2 ([hutchings2016beyond, Example 1.12]).
Let be a convex toric domain defined by a nice function . Let be a nondegenerate contact structure obtained by perturbing up to sufficiently large action. Then the orbit sets with ECH index are classified as follows.
: no orbit sets.
: and .
: , , and .
In general, the classification of orbit set generators, up to larger ECH index and action, provides a combinatorial model to compute the sequence of ECH capacities of a convex toric domain using the following formula.
Lemma 2.3 ([hutchings2016beyond, Lemma 5.6]).
For a convex toric domain and a nonnegative integer ,
In particular, the equalities claimed in Proposition 1.7 hold. Moreover, one can deduce the following lemma which we will use to rule out breaking.
For a convex toric domain , orbit sets with ECH index have action .
3 Ruling out breaking
3.1 Completed symplectic cobordisms
Let be closed contact –dimensional manifolds. A compact symplectic cobordism from to is a compact symplectic manifold with boundary such that .
Given a compact symplectic cobordism , one can find neighborhoods of and of in , and symplectomorphisms
where denotes the coordinate on and . Using these identifications, we can complete the compact symplectic cobordism by adding cylindrical ends and to obtain the completed symplectic cobordism
In accordance with [bourgeois2003compactness], we restrict the class of almost complex structures on a completed cobordism as follows. An almost complex structure on a completed symplectic cobordism as above is called compatible (in [bourgeois2003compactness], the authors use the term adjusted) if:
On and , the almost complex structure is –invariant, maps (the direction) to , and maps to itself compatibly with .
On the compact symplectic cobordism , the almost complex structure is tamed by .
Call the set of all such compatible almost complex structures on .
Choose a compatible almost complex structure on and a let be a compact Riemann surface. We will consider curves that are –holomorphic, i.e. , and have positive ends at corresponding to the punctures , and negative ends at corresponding to the punctures . Denote by the space of such –holomorphic curves modulo reparametrizations of the domain .
Recall that a positive end of at means a puncture, near which, is asymptotic to . More specifically, that means there is a choice of coordinates on a neighborhood of the puncture, with and such that and . Similarly, a negative end at is a puncture, near which, is asymptotic to . More specifically, that means there is a choice of coordinates on a neighborhood of the puncture, with and such that and .
Given a –holomorphic curve as above, define the Fredholm index of by
where is a trivialization of over that is symplectic with respect to , is the Euler characteristic of , denotes the relative first Chern class, and is the Conley–Zehnder index with respect to , as before. The significance of the Fredholm index is that for a generic choice of compatible almost complex structure and for a somewhere–injective –holomorphic curve , the moduli space is a manifold of dimension near . See [wendl2016lectures, §6] for more details.
3.2 Moduli spaces
Let and be two convex toric domains defined by nice functions and , respectively. Also, let be a symplectic embedding. The manifold is a compact symplectic cobordism from to , where denotes the standard Liouville form on .
Following the explanation in Remark 2.1, perturb the boundary components and of in such a way that the Liouville form restricts to nondegenerate contact forms and on and , respectively. Add cylindrical ends to and call the completed symplectic cobordism.
To clean up notation, call the embedded Reeb orbit on , and call the embedded Reeb orbit on . Recall that and .
For a given almost complex structure , define to be the moduli space of –holomorphic cylinders such that has a positive end at and a negative end at , modulo translation and rotations of the domain .
All such –holomorphic cylinders have Fredholm index and the automatic transversality result in Lemma 3.1 below implies that is a –dimensional manifold for any choice of . Moreover, can be compactified with broken holomorphic curves using the SFT compactness theorem, [bourgeois2003compactness, Theorem 10.2], since all the –holomorphic cylinders in have the same asymptotics.
3.3 Automatic transversality
A much more general automatic transversality result than the one we need to use is proven by Wendl in [wendl2008automatic]. In the language employed in this paper, the particular case that we need to use is stated as follows. See also [hutchings2014cylindrical, Lemma 4.1] for a very similar statement and proof in the case of symplectizations.
Let be a completed symplectic cobordism and let be an immersed –holomorphic curve that has asymptotic ends to Reeb orbits. Let denote the normal bundle to in and
denote the normal linearized operator of . Also let denote the number of ends of at positive hyperbolic orbits. If
then is surjective, i.e. the moduli space of –holomorphic curves near is a manifold that is cut out transversely and has dimension .
Note that there are no genericity assumptions on the almost complex structure in Lemma 3.1. Also, the result applies to the –holomorphic cylinders in since they have ends only at elliptic Reeb orbits and the adjunction formula introduced below in (8) implies that they are embedded. Hence is cut out transversely, for any choice of compatible almost complex structure .
3.4 Ruling out breaking
In this section, we study the possible boundary of the union , where is a smooth parametrized family of compatible almost complex structures. We prove that, assuming the bounds in the hypothesis of Theorem 1.8, a sequence of cylinders in cannot converge to a broken holomorphic building with multiple levels.
Assume and are convex toric domains satisfying the bounds in the hypothesis of Theorem 1.8. Let be a sequence of symplectic embeddings, –converging to . Let be a sequence of compatible almost complex structures converging to . Let . Then the sequence cannot converge in the sense of [bourgeois2003compactness] to a –holomorphic building with more than one level.
In general, if there exists a –holomorphic curve from the orbit set to the orbit set , then . Assume that, in the limit, the cylinders break into a –holomorphic building . Assume that is the orbit set at which the level has negative ends. Then . Note first that is the lowest action of an orbit set in . This means that lives in the cobordism level. Secondly, the assumption translates to
where , , and are the Reeb orbits on