Compactness Results for –Holomorphic Maps
–holomorphic maps are a parameter version of –holomorphic maps into contact manifolds. They have arisen in efforts to prove the existence of higher–genus holomorphic open book decompositions, the existence of finite energy foliations and the Weinstein conjecture [ACH05], as well as in folded holomorphic maps [vB07]. For all these applications it is essential to understand the compactness properties of the space of –holomorphic maps.
We prove that the space of –holomorphic maps with bounded periods into a manifold with stable Hamiltonian structure possesses a natural compactification. Limits of smooth maps are neck-nodal maps, i.e. their domains can be pictured as nodal domains where the node is replaced by a finite cylinder that converges to a twisted cylinder over a closed characteristic or a finite length characteristic flow line. We show by examples that compactness fails without the condition on the periods, and we give topological conditions that ensure compactness.
The theory of –holomorphic curves has become an indispensable tool for symplectic and contact geometry and topology. However in many potential applications for –holomorphic curves the index for the curves of interest turns out to be negative, or have dimension too low. In particular, this happens when one tries to foliate a 4–manifold, or the symplectization of a contact 3–manifold, by –holomorphic surfaces of genus . In this situation one considers embedded curves with trivial normal bundle, so the index is , when one would like the index to be 2, the dimension of the leaf space of the foliation. To remedy this Abbas Cieliebak and Hofer suggested using families of –holomorphic curves parameterized by [ACH05]. The same parameter space is also needed for index reasons in [vB07]. These family version of –holomorphic maps are called –holomorphic maps.
is a stable Hamiltonian structure if and satisfy
The stable Hamiltonian structure induces as splitting , where
is called the characteristic foliation and the section of defined by is called the characteristic vector field. is a symplectic vector bundle. In the case that , is called a contact form, a contact structure and the Reeb vector field.
Let be the set of almost complex structures on that are compatible with , i.e.
Often we will refer to a stable Hamiltonian structure as including a choice of . This gives rise to a metric on .
Definition 1.2 (–Holomorphic Maps).
Let be a punctured Riemann surface. A map is called –holomorphic if
This is a system of elliptic differential equations, a (first order) Cauchy-Riemann type equation in the directions and a (second order) Poisson equation in the direction. The last equation demands that the periods of vanish at the punctures.
–holomorphic maps can be viewed as –holomorphic maps with parameter space in the following way. Let of harmonic 1–forms on , i.e.
For an –holomorphic map there is a unique , and a function that is unique up to addition of a constant, so that
Given an –holomorphic map we will often make implicit use of the splitting of closed 1–forms given in Equation (1.5) without mention. The pair
is called the canonical lift of to the symplectization. It is unique up to translation in the –factor. With this notation the map is –holomorphic if and only if
where is the canonical –invariant almost complex structure on the symplectization and is the –part of the complexification of the space of harmonic 1–forms on viewed as taking values in the trivial complex subbundle . In particular, every –holomorphic map is also –holomorphic.
The important feature of the space is that it gives a complement of the coexact 1–forms in the space of coclosed 1–forms. Sometimes it is convenient to choose a different complement with some prescribed properties and consider lifts to the symplectization w.r.t. , i.e. maps with .
Locally all closed forms are exact, so –holomorphic maps inherit all local properties of –holomorphic maps. By standard theory (see e.g. [HWZ96]) –holomorphic maps that satisfy certain energy assumptions limit to closed characteristics at the punctures and the maps extend to a continuous map from the radial compactification of .
In order for –holomorphic maps to be useful for applications in symplectic and contact geometry, it is important for the moduli space of –holomorphic maps to possess a natural compactification. The non–compactness of this parameter space is the source for the compactness issues of the space of maps.
The idea of using a parameter space to change the index of an equation to fit applications has a long history, and usually requires a delicate analysis of solutions. For example, Junho Lee [Lee04] considered families of –holomorphic maps into Kähler manifolds with non–compact parameter space given by harmonic 1–forms on the target. While in that case the space of maps is in general not compact, he was able to show that in certain interesting cases the space of maps stay in a compact subset of the parameter space and thus are compact.
We show that the situation for –holomorphic maps is quite similar. The space of maps is in general not compact (Theorem 2.9), and boundedness of the “periods” (Definition 2.7) is a necessary and sufficient condition on a sequence of smooth –holomorphic maps to have a convergent subsequence (Theorem 2.8, see Figure 1). This condition is automatically satisfied in situations arising for important applications (Theorem 4.2). In a sequel to this we are applying these results to nicely embedded –holomorphic maps and open book decompositions [vB09].
2 Main Results
In order to understand the precise compactness statement we briefly survey some related compactness results in the literature.
For harmonic maps, bubble tree convergence in a fixed homology class with fixed domain complex structure was proved by Parker [Par96]. There it was also observed that a similar result with varying complex structure on the domain does not hold due to loss of control over the “neck” regions.
Chen and Tian proved [CT99] that compactness for energy minimizing finite energy harmonic maps with domain complex structure converging in can be achieved if one fixes the homotopy class of maps, rather than just homology. Then the neck maps converge to finite length geodesics.
-holomorphic maps into contact manifolds do not have uniform (or even finite) -energy bounds. To overcome this Hofer, Wysocki and Zehnder ([HWZ96]) tailored a suitable notion of energy that is a homological invariant and guarantees bubble tree convergence where nodes (and punctures) can “open up” to wrap closed characteristics. In [BEH03] this has been extended to the case of manifolds with stable Hamiltonian structure as targets and allowing certain degenerations of the almost complex structure on the target.
In the case of -holomorphic maps there are again no uniform -energy bounds. Roughly speaking, the -holomorphic map equation into a - dimensional manifold with stable Hamiltonian structure is a mixture of a -dimensional -holomorphic map equation in the almost contact planes and a 1-dimensional harmonic map equation in the characteristic direction. This dual nature is reflected in the compactness statement as neck maps converge to “twisted cylinders” over closed characteristics or finite length characteristic flow lines.
In order to account for the possibility of necks converging to characteristic flow lines we make the following definition for the space of domains.
Fix a genus reference surface with boundary components. Denote the surface obtained by collapsing each boundary component to a point by . A neck-nodal domain of genus with boundary components and necks is given by a map
to a nodal curve with nodes and marked points such that
each boundary component is mapped to a marked point in , called a puncture,
there are embedded loops with pairwise disjoint tubular neighborhoods bounded away from each other and the boundary of and not containing any of the marked points so that each neck domain
maps to a distinct node of , and
is a diffeomorphism from the smooth part of
onto the curve obtained from by removing the punctures and nodes.
Thus induces a complex structure on the punctured surface . We denote the space of neck-nodal domains of genus with boundary components and marked points modulo diffeomorphisms of preserving the boundary components by .
There is an obvious bijection
to the space of decorated nodal surfaces defined in [BEH03], and we endow with the same topology as .
The neck domains don’t carry a well defined conformal structure. Intuitively they should be viewed as flat cylinders with infinitesimal circumference.
A neck map is a map of the form
where is a flow line of the characteristic vector field and . is called the period of the neck and is called the twist of the neck.
Note that if then is necessarily a -periodic orbit.
Each neck region defines an element , where is as always defined to be the surface obtained from by collapsing the boundary components.
Let be a neck-nodal domain with neck domains . Then the collection of neck maps has minimal twist if whenever for some index set , then there exists a collection of non–negative real numbers , with so that
where is the twist of .
We need one more definition ensuring that maps from a singular domain can be lifted to the symplectization.
A neck region is called non–separating if both boundary components of are adjacent to the same connected component of the smooth part of .
An –holomorphic map from the smooth part is called exact if it lifts to a map to the symplectization so that the –component of the lift extends continuously by a constant over the non–separating components of the necks.
We are now prepared for the definition of neck–nodal maps.
An -holomorphic map from a neck-nodal domain with neck domains into is a continuous map from into that restricts to an exact -holomorphic map from the smooth part into , and a minimal twist neck map on the neck domains.
This definition allows for –holomorphic maps in the compactification with qualitatively different behavior from –holomorphic maps. This is necessary as such maps occur in examples (see the end of Section 4). If a neck region is homologically trivial in , then a minimal twist neck map has vanishing twist on . This means that –holomorphic maps exhibit “zero distance bubbling”, just like in the –holomorphic and harmonic map case.
To illustrate the meaning of the minimal twist condition further let be the genus of and denote the genus of the normalization of corresponding nodal curve by . Let be the number of neck regions and the number of independent relations of the neck regions in . Then . In this equation is half the number of harmonic 1–forms that are lost in the singular domain. Half of the lost harmonic 1–forms are fixed as the periods of necks, and the other half is encoded in the n twist parameters of the necks satisfying the relations.
There are several notions of energy that are important for –holomorphic maps. With the space of probability measures on the real line we make use of the following standard definition for –holomorphic maps.
Let be the space of smooth probability measures on the real line. The –energy and –energy are
The integrands are pointwise non–negative functions on and both energies are invariants of the relative homology class of the map . By the definition of stable Hamiltonian structure Definition 1.1 there exists a constant so that
for any –holomorphic map from a Riemann surface, possibly with boundary, . In particular, finite –energy of an –holomorphic map implies that is also finite for any subdomain .
We will see that in order to prove compactness of a family of –holomorphic maps it is necessary and sufficient that the parameters stay in a compact subset of . Compactness is to be understood with respect to the topology induced by the period map (on a basis of ). Here it is essential that the harmonic 1–form in question is defined as the harmonic part of , and not of if one wishes to consider sequences of complex structures converging to the boundary of .
We wish to find a useful criterion to check if the periods of the harmonic parts of of a sequence of –holomorphic maps remains bounded. To this end, we associate to a canonical family of curves along which we will evaluate the integrals of . It turns out that one-cylinder Strebel differentials are a convenient tool for this. We quickly outline the relevant portions of the theory, for more details we refer the interested reader to [Str84].
If has genus 0, then is trivial, so every –holomorphic map from a domain of genus 0 is automatically –holomorphic. Since the compactness properties of –holomorphic maps is already well understood we will restrict our attention to domains of genus .
A holomorphic quadratic differential is a tensor, locally in complex coordinates , given as , where is a holomorphic function. defines a singular Euclidean metric on with finitely many singular points corresponding to the zeros of . determines a pair of transverse measured foliations and called the horizontal and vertical foliations given by the preimages of the real and imaginary axes under , respectively. Near a singular point of of order , is given in local coordinates as . The union of the leaves both beginning and ending at a critical point is called the critical graph .
Given a non–separating simple closed curve on , there exists a holomorphic quadratic differential , called the Strebel differential, so that its horizontal foliation has closed leaves in the free homotopy class . Denote the set of such Strebel differential associated with and by .
The complement of the critical graph is a metric (w.r.t. ) cylinder . If has genus 1, then there is no critical graph and we use one regular leaf for . For simplicity we normalize so that has height 1. For details see [Str84], Theorem 21.1. Let be a parametrization the closed leaves of the horizontal foliation .
Armed with this definition we are ready to define the periods of –holomorphic maps.
Let be a non–separating simple closed curve in and a 1–form on . Then the period of along are
For an –holomorphic map define the period
We say that a family of maps has bounded periods if there exists a collection of simple closed curves that form a basis of so that the associated periods are uniformly bounded.
The definition of the periods along a curve is somewhat abstract. Intuitively, we think of the periods as the periods of the non–closed form . The one–cylinder Strebel differentials allow us to define the periods in a way that is invariant under the gauge action by diffeomorphisms, and independent of the choice of conformal metric on the domain. It turns out that the periods are essentially given by the periods of the harmonic part of the co–closed form (see Lemma 3.2).
Bounded periods are a necessary condition for any meaningful compactness result, as the a sequence of maps with unbounded periods has unbounded diameter in the image. The following theorem shows that the converse is also true, i.e. that bounded periods lead to compactness.
Let be a stable Hamiltonian structure so that all periodic orbits are Morse or Morse-Bott. The space of smooth -holomorphic maps into with uniformly bounded and –energies with uniformly bounded periods has compact closure in the space of neck–nodal –holomorphic maps.
In Section 4 we give examples of topological conditions that guarantee that the periods of families of maps are uniformly bounded, leading to compact moduli spaces. Another such condition guaranteeing bounded periods is given in a follow–up paper [vB09] when considering nicely embedded maps.
The condition on the periods is not vacuous as the following result shows.
Let be the twice–punctured standard torus and let be equipped with the standard contact form and complex structure. There exists a smooth family of –holomorphic maps parametrized by so that a sequence , has a convergent subsequence if and only if has a convergent subsequence in as . The width of the image becomes unbounded as .
The existence of non–compact smooth families of maps stands in stark contrast to the case of –holomorphic maps and destroys any hope for a general compactness theorem for –holomorphic maps.
3 Bounded Periods and Compactness
In this section we prove Theorem 2.8. In the first part, we show that the requirement of bounded periods of the maps leads to bounded periods of the harmonic 1–forms, and that every sequence with bounded periods possesses a subsequence that converges on compact subsets of the complement of the necks.
In the following subsection we investigate the convergence of the maps on the necks, where the requirement of bounded periods will lead to neck maps with minimal twist.
Putting these results together we then proceed to prove Theorem 2.8.
First we explain the metrics we are using on the domains. Following the construction in Section 4 of [IP04] we choose a family of metrics on the space of the domains , of –holomorphic maps that comes from a metric on the universal curve on the thick part of the domain and is given by the cylindrical metric on (a cofinite subset of) the thin part. More specifically we adopt the definition of the weight function and work with the metric , where is the restriction of a Riemannian metric on the universal curve. In particular, near a puncture we have local coordinates with metric , and , and , and on a neck cylinder we have the same flat metric and . Given a sequence of maps and conformal structures we may adjust the space every time we rescale (a finite number of times) by adding marked points as needed, and we will adjust the metrics accordingly without making explicit mention of this.
We refer to this metric as the cylindrical metric and will use it throughout for estimates. For the final statement of the compactness theorem we will however use a different metric, namely the non–conformal metric where the cylinders of the thin part are rescaled along the height of the cylinders to , where the scaling function depends on the asymptotic approach of the maps to a closed characteristic. This metric extends to a smooth metric on the space of neck–nodal domains, and the convergence results in the compactness statement are to be understood with respect to this metric.
Let be a an –holomorphic map and let be the sum of the energy of and be the of the sum of the absolute values of the periods of at the punctures. Then, for any non–trivial free homotopy-separating simple closed curve , the periods , where is the co-exact part of , i.e. .
Let be a foliation with compact leaves given by a Strebel differential with ring domain of height 1 associated with , and let denote it’s singular leaf. So .
Here we interpret integrals on a leaf containing a puncture in the sense of Cauchy. Then for any
where is the sum of the absolute value of the periods, so
Since this is true for any Strebel differential and any leaf we conclude that
We immediately obtain the following results.
Let be a sequence of –holomorphic maps with fixed asymptotics and uniformly bounded –energy. Then has uniformly bounded periods if and only if the periods of are uniformly bounded.
Let be a sequence of –holomorphic maps so that the periods of are uniformly bounded. Then, after finitely many rescalings, there is a subsequence so that is uniformly bounded.
By Lemma A.3 we see that is uniformly bounded. The result now follows from the standard bubbling off analysis. ∎
For an –holomorphic map from a cylinder we define the twist
and the average twist
The twist and the average twist only depend on and are independent of , since
In particular, the twist of a neck–region is uniformly bounded in terms of the periods of by Lemma A.2, and the relative twist is bounded in terms of .
Let be a sequence of –holomorphic maps with and uniformly bounded –energy and –energy and uniformly bounded periods.
Then there exists a constant and a subsequence so that with
and the twist of all neck–maps is uniformly bounded.
By Lemma 3.2 we see that the periods of are uniformly bounded. Then Lemma 3.3 shows that we can, after finitely many rescalings, choose a subsequence so that is uniformly bounded. By Lemma A.3 we see that the sup norm of is also uniformly bounded, and thus and must also be uniformly bounded in the sup norm. The twists of the neck maps are uniformly bounded by Lemma A.2. ∎
3.1 Long Cylinders
To prove the compactness statement we need to understand the behavior of long –holomorphic cylinders with small –energy and uniformly bounded derivative, center action, and twist. We reduce the argument to the –holomorphic case discussed in [HWZ02]. The main difference between the and –holomorphic settings is that –holomorphic maps may have non–zero (uniformly bounded) twist, whereas –holomorphic maps have vanishing twist.
To reduce the question about –holomorphic cylinders to –holomorphic cylinders, let denote the time– characteristic flow. The bundle map is an isomorphism preserving the splitting . Given an –holomorphic map with average twist let be given by
and define the 1–parameter family of almost complex structures
Let be the domain dependent almost complex structure defined by . Then is –holomorphic, that is is coexact and
Now suppose the twists of a family of –holomorphic maps are uniformly bounded by . Then for each , the family , of almost complex structures varies in the compact set
independent of how large is.
Before we proceed we need the following definition.
The period spectrum of is
For any the period gap w.r.t is the largest number so that
Let so that all closed characteristics of period are non–degenerate. Let be the period gap between closed characteristics of period as in Equation (3.9). Let be smaller than the lowest eigenvalue of any asymptotic operator governing the transverse approach to any closed characteristic of period .
Fix satisfying and . Then for every there exists a constant so that the following holds.
For every and every –holomorphic cylinder
satisfying and gradient and twist bounded by and center action satisfies there exist a characteristic flow line so that
Consider the –holomorphic map
and note that and lifts to a finite –energy –holomorphic map into the symplectization. Moreover the energy is a–priori bounded in terms of the center action and the –energy.
To prove the theorem we need to show that there exists so that
But this follows directly from Theorems 1.2 and 1.3 of [HWZ02]. ∎
3.2 Proof of Theorem 2.8
Before we proof Theorem 2.8 we observe some relations among the neck lengths.
Let be a sequence of complex structures on and a sequence of harmonic 1–forms with converging period integrals. Then the twists of on each neck converge to real numbers , and there exists a subsequence so that whenever for some index set there exist non–negative real numbers with so that .
In particular, homologically trivial necks have vanishing twist.
Consider on the necks . With the center loops of each neck. The periods of on satisfy
Let be an index set so that and define, for
The twist of on the neck satisfy,
Set , which exists by assumption, and choose a subsequence so that exists. For each we have
Since there are only finitely many index sets so that there exists a subsequence so that the Lemma holds true. ∎
Proof of Theorem 2.8.
Let be a sequence of smooth –holomorphic maps with bounded and –energies and periods bounded by . We need to show that there exists a subsequence that converges to a neck–nodal –holomorphic map.
By Theorem 3.5 we may pass to a subsequence so that and , and the relative twists of the neck maps are uniformly bounded. By elliptic regularity and Arzela–Ascoli we extract a convergent (in ) on the thick part of .
By Lemma 3.8 we may extract a subsequence so that the twists of on the necks are bounded, convergent, and the twists are minimal.
By our assumption we have that the center action of each neck is bounded by some constant , so by Lemma 3.7 we see that there exists constants and a characteristic flow line so that and for . Since the gradient of is uniformly bounded on we may assume, by adjusting the constant in the above formulas, that .
Set . We may split a neck region into regions
We similarly split up the cylinder into regions
and define the piecewise diffeomorphism via the diffeomorphisms
Define the map and consider the restrictions to the subdomains and . Then, with coordinates on and remembering that
which is uniformly bounded by and converges to zero as and . Similarly
which is also uniformly bounded and converges to zero for and . So there exists a reparametrization and a subsequence so that converges to an –holomorphic map from the smooth part of the neck–nodal domain . Note that since the average twist on each neck converges to zero uniformly we have that uniformly, so in the limit condition 1.4 of Definition 1.2 is also satisfied along the necks which are now punctures for .
Similarly, we compute
which converges to zero uniformly, so a subsequence of converges uniformly to a neck map.
Using the diffeomorphisms we reinterpret our sequence of maps as maps from a fixed reference surface by gluing the thick part to the cylinders . After passing to a subsequence the resulting domains with their induced complex structure converge to a neck–nodal domain, and the resulting maps converge uniformly in to a minimal twist neck–nodal –holomorphic map .
Standard arguments show that the canonical lifts to the symplectization also converge, so is exact. ∎
In fact, it is not hard to extend these results to make the convergence piecewise smooth, so that the convergence is on the neck regions and the smooth part of the domain.
4 –Invariant Stable Hamiltonian Manifolds
In this section we consider circle–invariant stable Hamiltonian manifolds of any (odd) dimension. We give topological conditions under which the periods of families of –holomorphic maps are always uniformly bounded, and thus obtain compact moduli spaces of maps. –holomorphic maps into circle–invariant manifolds are needed for applications to folded holomorphic maps [vB07]. They also allow for the explicit construction of examples highlighting the features of the compactness theorem, which we give at the end of this section, as well as the counterexample to a general compactness theorem which is constructed in Section 5.
An stable Hamiltonian manifold is called –invariant if the characteristic flow defines a free –action that preserves .
Any circle–invariant manifold is an –bundle over a symplectic manifold with projection so that