Symplectic and contact differential graded algebras

# Symplectic and contact differential graded algebras

Tobias Ekholm Uppsala University, Box 480, 751 06 Uppsala, Sweden, and Institute Mittag-Leffler, Aurav 17, 182 60 Djursholm, Sweden  and  Alexandru Oancea Sorbonne Universités, UPMC Univ Paris 06, UMR 7586, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Case 247, 4 place Jussieu, F-75005, Paris, France
June 2, 2015
###### Abstract.

We define Hamiltonian simplex differential graded algebras (DGA) with differentials that deform the high energy symplectic homology differential and wrapped Floer homology differential in the cases of closed and open strings in a Weinstein manifold, respectively. The order term in the differential is induced by varying natural degree co-products over an -simplex, where the operations near the boundary of the simplex are trivial. We show that the Hamiltonian simplex DGA is quasi-isomorphic to the (non-equivariant) contact homology algebra and to the Legendrian homology algebra of the ideal boundary in the closed and open string cases, respectively.

T.E. partially funded by the Knut and Alice Wallenberg Foundation as a Wallenberg scholar and by the Swedish Research Council, 2012-2365
A.O. partially funded by the European Research Council, StG-259118-STEIN

## 1. Introduction

Let be a Weinstein manifold and let be an exact Lagrangian submanifold. (We use the terminology of [15] for Weinstein manifolds, cobordisms etc. throughout the paper.) Assume that is cylindrical at infinity meaning that outside a compact set looks like , where is a contact manifold, a Legendrian submanifold, and the Liouville form on is the symplectization form , for a contact form on and the standard coordinate in .

There are a number of Floer homological theories associated to this geometric situation. For example there is symplectic homology which can be defined [40, 37, 11] using a time-dependent Hamiltonian , which is a small perturbation of a time independent Hamiltonian that equals a small positive constant in the compact part of and is linearly increasing of certain slope in the coordinate in the cylindrical end at infinity, and then taking a certain limit over increasing slopes. The chain complex underlying is denoted and is generated by the -periodic orbits of the Hamiltonian vector field of , graded by their Conley-Zehnder indices. These fall into two classes: low energy orbits in the compact part of and (reparameterizations of) Reeb orbits of in the region in the end where increases from a function that is close to zero to a function of linear growth. The differential counts Floer holomorphic cylinders interpolating between the orbits. These are solutions , of the Floer equation

 (1.1) (du−XH⊗dt)0,1=0,

where is a standard complex coordinate and the complex anti-linear part is taken with respect to a chosen adapted almost complex structure on . The -periodic orbits of are closed loops that are critical points of an action functional, and cylinders solving (1.1) are similar to instantons that capture the effect of tunneling between critical points. Because of this and analogies with (topological) string theory, we say that symplectic homology is a theory of closed strings.

The open string analogue of is a corresponding theory for paths with endpoints in the Lagrangian submanifold . It is called the wrapped Floer homology of and here denoted . Its underlying chain complex is generated by Hamiltonian time chords that begin and end on , graded by a Maslov index. Again these fall into two classes: high energy chords that correspond to Reeb chords of the ideal Legendrian boundary of and low energy chords that correspond to critical points of restricted to . The differential on counts Floer holomorphic strips with boundary on interpolating between Hamiltonian chords, i.e. solutions

 u:(R×I,∂(R×I))→(X,L)

of (1.1).

We will also consider a mixed version of open and closed strings. The graded vector space underlying the chain complex is simply , and the differential has the following matrix form with respect to this decomposition (subscripts “” and “” refer to closed and open, respectively):

 d1=(dccdoc0doo).

Here and are the differentials on and , respectively, and is a chain map of degree . Each of these three maps counts solutions of (1.1) on a Riemann surface with two punctures, one positive regarded as input, and one negative regarded as output. For the underlying Riemann surface is the cylinder, for the underlying Riemann surface is the strip, and for the underlying Riemann surface is the cylinder with a slit at (or equivalently, a disk with two boundary punctures, a sphere with two interior punctures, and a disk with positive boundary puncture and negative interior puncture). We will denote the corresponding homology .

In order to count the curves in the differential over integers we use index bundles to orient solution spaces and for that we assume that the pair is relatively spin, see [25]. As the differential counts Floer-holomorphic curves, it respects the energy filtration and the subspace generated by the low energy chords and orbits is a subcomplex. We denote the corresponding high energy quotient and its homology . We define similarly , , , and .

In the context of Floer homology, the cylinders and strips above are the most basic Riemann surfaces, and it is well-known that more complicated Riemann surfaces can be included in the theory as follows, see [37, 34]. Pick a family of 1-forms with values in Hamiltonian vector fields on over the appropriate Deligne-Mumford space of domains and count rigid solutions of the Floer equation

 (1.2) (du−B)0,1=0,

where in cylindrical coordinates near the punctures of . The resulting operation descends to homology as a consequence of gluing and Gromov-Floer compactness. A key condition for solutions of (1.2) to have relevant compactness properties is that is required to be non-positive in the following sense. For each we get a 1-form on with values in , , where , is a family of time dependent Hamiltonian functions parameterized by and is a 1-form on . The non-positivity condition is then that the 2-form associated to , , is a non-positive multiple of the area form on for each .

The most important such operations on are the BV-operator and the pair-of-pants product. The BV-operator corresponds to solutions of a parameterized Floer equation analogous to (1.1) which twists the cylinder one full turn. The pair-of-pants product corresponds to a sphere with two positive and one negative puncture and restricts to the cup product on the ordinary cohomology of , which here appears as the low energy part of . Analogously, on the product corresponding to the disk with two positive and one negative boundary puncture restricts to the cup product on the cohomology of , and the disk with one positive interior puncture and two boundary punctures of opposite signs expresses as a module over .

Seidel [37] showed that such operations are often trivial on . Basic examples of this phenomenon are the operations given by disks and spheres with one positive and negative punctures. By pinching the -form in (1.2) in the cylindrical end at one of the negative punctures, it follows that up to homotopy factors through the low energy part of the complex . In particular, on the high energy quotient the operation is trivial if the -form is pinched near at least one negative puncture.

The starting point for this paper is to study operations that are associated to natural families of forms that interpolate between all ways of pinching near negative punctures. More precisely, for disks and spheres with one positive and negative punctures, we take in (1.2) to have the form in the cylindrical end, with coordinate in for open strings and in for closed strings, near the puncture. Here is a positive function with a minimal value called weight. By Stokes’ theorem, in order for to satisfy the non-positivity condition, the sum of weights at the negative ends must be greater than the weight at the positive end. Thus the choice of -form is effectively parameterized by an -simplex and the equation (1.2) associated to a form which lies in a small neighborhood of the boundary of the simplex, where at least one weight is very small, has no solutions with all negative punctures at high energy chords or orbits. The operation is then defined by counting rigid solutions of (1.2) where varies over the simplex bundle. Equivalently, we count solutions with only high energy asymptotes in the class dual to the fundamental class of the sphere bundle over Deligne-Mumford space obtained as the quotient space after fiberwise identification of the boundary of the simplex to a point. In particular, curves contributing to have formal dimension .

Our first result says that the operations combine to give a DGA differential. The Hamiltonian simplex DGA is the unital algebra generated by the generators of with grading shifted down by , where orbits sign commute with orbits and chords but where chords do not commute. Let be the map defined on generators by

 db=d1b+d2b+⋯+dmb+⋯,

and extend it by the Leibniz rule.

###### Theorem 1.1.

The map is a differential, , and the homotopy type of the Hamiltonian simplex DGA depends only on . Furthermore, is functorial in the following sense. If , if is a Weinstein cobordism with negative end , and if denotes the Weinstein manifold obtained by gluing to , then there is a DGA map

 ΦX10:SC+(X1,L1)→SC+(X0,L0),

and the homotopy class of this map is an invariant of up to Weinstein homotopy.

If in Theorem 1.1 then we get a Hamiltonian simplex DGA generated by high energy Hamiltonian orbits. This DGA is (graded) commutative. Also, the quotient of by the ideal generated by orbits is a Hamiltonian simplex DGA generated by high energy chords of . We write for the homology DGA of , and use the notation and with a similar meaning. If is the cotangent bundle of a manifold then is isomorphic to the homology of the free loop space of , see [39, 36, 2, 3], and the counterpart of in string topology is non-trivial, see [26].

Our second result expresses in terms of the ideal boundary . Recall that the usual contact homology DGA is generated by closed Reeb orbits in and by Reeb chords with endpoints on , see [24]. Here we use the differential that is naturally augmented by rigid once-punctured spheres in and by rigid once-boundary punctured disks in with boundary in . (In the terminology of [7] the differential counts anchored spheres and disks). In [10] a non-equivariant version of linearized orbit contact homology was introduced. In Section 6 we extend this construction and define a non-equivariant DGA that we call , which is generated by decorated Reeb orbits and by Reeb chords. We give two definitions of the differential on , one using Morse-Bott curves and one using curves holomorphic with respect to a domain dependent almost complex structure. In analogy with the algebras considered above we write for the subalgebra generated by decorated orbits and for the quotient by the ideal generated by decorated orbits.

In Sections 2.6 and 6.1 we introduce a continuous -parameter deformation of the simplex family of -forms that turns off the Hamiltonian term in (1.2) by sliding its support to the negative end in the domains of the curves and that leads to the following result.

###### Theorem 1.2.

The deformation that turns the Hamiltonian term off gives rise to a DGA map

 Φ:A(Y,Λ)→SC+(X,L).

The map is a quasi-isomorphism that takes the orbit subalgebra quasi-isomorphically to the orbit subalgebra . Furthermore, it descends to the quotient and maps it to as a quasi-isomorphism.

The usual (equivariant) contact homology DGA is also quasi-isomorphic to a Hamiltonian simplex DGA that corresponds to a version of symplectic homology defined by a time independent Hamiltonian, see Theorem 6.3. For the corresponding result on the linear level, see [12].

###### Remark 1.3.

As is well-known, the constructions of the DGAs and , of the orbit augmentation induced by , and of symplectic homology for time independent Hamiltonians require the use of abstract perturbations for the pseudo-holomorphic curve equation in a manifold with cylindrical end. This is an area where much current research is being done and there are several approaches, some of an analytical character, see e.g. [28, 29], others of more algebraic topological flavor, see e.g. [33], and others of more geometric flavor, see e.g. [25]. Here we will not enter into the details of this problem but merely assume such a perturbation scheme has been fixed. Our results are independent of the nature of the perturbation scheme and use only the weakest form of it that allows us to count rigid curves over the rationals. In fact, in this spirit, Theorem 1.2 can be interpreted as an alternative definition of the (non-equivariant) contact homology DGA that does not involve abstract perturbations.

Theorem 1.2 relates Symplectic Field Theory (SFT) and Hamiltonian Floer Theory. On the linear level the relation is rather direct, see [10], but not for the SFT DGA. The first candidate for a counterpart on the Hamiltonian Floer side collects the standard co-products to a DGA-differential, but that DGA is trivial by pinching. To see that recall the sphere bundle over Deligne-Mumford space obtained by identifying the boundary points in each fiber of the simplex bundle. The co-product DGA then corresponds to counting curves lying over the homology class of a point in each fiber, but that point can be chosen as the base point where all operations are trivial. The object that is actually isomorphic to the SFT DGA is the Hamiltonian simplex DGA related to the fundamental class of the spherization of the simplex bundle.

In light of this, the following picture of the relation between Hamiltonian Floer Theory and SFT emerges. The Hamiltonian Floer Theory holomorphic curves solve a Cauchy-Riemann equation with Hamiltonian -order term chosen consistently over Deligne-Mumford space. These curves are less symmetric than their counterparts in SFT, which are defined without additional -order term. Accordingly, the moduli spaces of Hamiltonian Floer Theory have more structure and carry natural actions, e.g. of scaling simplices and the framed little disk operad, see Section 7. The SFT moduli spaces are in a sense homotopic to certain essential strata inside the Hamiltonian Floer Theory moduli spaces, and the structure and operations that they carry are intimately related to the natural actions mentioned. From this perspective, this paper studies the action given by scaling simplices in the most basic case of higher co-products.

We end the introduction by a comparison between our constructions and other well-known constructions in Floer theory. In the case of open strings, the differential can be thought of as a sequence of operations on the vector space . These operations define the structure of an -coalgebra on (with grading shifted down by one) and is the cobar construction for this -coalgebra. This point of view is dual to that of the Fukaya category, in which the primary objects of interest are -algebras. In the Fukaya category setting algebraic invariants are obtained by applying (variants of) the Hochschild homology functor. In the DGA setting invariants are obtained more directly, as the homology of the Hamiltonian simplex DGA.

#### Acknowledgements

Both authors would like to thank the organizers of the Gökova 20th Geometry and Topology Conference held in May 2013 for an inspiring meeting, during which the first ideas related to this paper crystallized. An early January 2013 discussion between the second author and Mohammed Abouzaid about the symplectic homology coproduct was also important. Part of this work was carried out while A.O. visited the Simons Center for Geometry and Physics at Stony Brook in the summer 2014.

## 2. Simplex bundles over Deligne-Mumford space, splitting compatibility, and 1-forms

The Floer theories we study use holomorphic maps of disks and spheres with one positive and several negative punctures. Configuration spaces for such maps naturally fiber over the corresponding Deligne-Mumford space that parameterizes their domains. In this section we endow the Deligne-Mumford space with additional structure needed to define the relevant solution spaces. More precisely, we parameterize -forms with non-positive exterior derivative by a simplex bundle over Deligne-Mumford space that respects certain restriction maps at several level curves in the boundary. We then combine these forms with a certain type of Hamiltonians to get non-positive forms with values in Hamiltonian vector fields, suitable as 0-order perturbations in the Floer equation.

### 2.1. Asymptotic markers and cylindrical ends

We will use punctured disks and spheres with a fixed choice of cylindrical end at each puncture. Here, a cylindrical end at a puncture is defined to be a biholomorphic identification of a neighborhood of that puncture with one of the following punctured model Riemann surfaces:

• Negative interior puncture:

 Z−=(−∞,0)×S1≈D2∖{0},

where is the unit disk in the complex plane.

• Positive interior puncture:

 Z+=(0,∞)×S1≈C∖¯D2.
• Negative boundary puncture:

 Σ−=(−∞,0)×[0,1]≈(D2∖{0})∩H,

where denotes the closed upper half plane.

• Positive boundary puncture:

 Σ+=(0,∞)×[0,1]≈(C∖¯D2)∩H.

Each of the above model surfaces has a canonical complex coordinate of the form . Here at all punctures, with or according to whether the puncture is positive or negative. At interior punctures, and at boundary punctures, .

The automorphism group of the cylindrical end at a boundary puncture is and the end is thus well-defined up to a contractible choice of automorphisms. For a positive or negative interior puncture, the corresponding automorphism group is . Thus the cylindrical end is well-defined up to a choice of automorphism in a space homotopy equivalent to . To remove the -ambiguity, we fix an asymptotic marker at the puncture, i.e. a tangent half-line at the puncture, and require that it corresponds to or to , , at positive or negative punctures, respectively. The cylindrical end at an interior puncture with asymptotic marker is then well-defined up to contractible choice.

We next consider various ways to induce asymptotic markers at interior punctures that we will eventually assemble into a coherent choice of asymptotic markers over the space of punctured spheres and disks. Consider first a disk with interior punctures and with a distinguished boundary puncture . Then determines an asymptotic marker at any interior puncture as follows. There is a unique holomorphic diffeomorphism with and . Define the asymptotic marker at in to correspond to the direction of the real line at , i.e. the direction given by the vector . See Figure 1.

Similarly, on a sphere , a distinguished interior puncture with asymptotic marker determines an asymptotic marker at any other interior puncture as follows. There is a holomorphic map taking to , to , and the asymptotic marker to the tangent vector of . We take the asymptotic marker at to correspond to the tangent vector of at under . See Figure 1.

For a more unified notation below we use the following somewhat involved convention for our spaces of disks and spheres. Let . For and , let denote the moduli space of disks with one positive boundary puncture, negative boundary punctures, and negative interior punctures. For and , let denote the moduli space of spheres with one positive interior puncture with asymptotic marker and negative interior punctures.

As explained above there are then, for both and , induced asymptotic markers at all the interior negative punctures of any element in . The space admits a natural compactification that consists of several level disks and spheres, see [8, §4] and also [30]. We introduce the following notation to describe the boundary. Consider a several level curve. We associate to it a downwards oriented rooted tree with one vertex for the positive puncture of each component of the several level curve and one edge for each one of the negative punctures of the components of the several level curves. See Figure 2 for examples. Here the root of the tree is the positive puncture of the top level curve and the edges attached to it are the edges of the negative punctures in the top level oriented away from the root. The definition of is inductive: the vertex of the positive puncture of a curve in the level is attached to the edge of the negative puncture of a curve in the level where it is attached. All edges of negative punctures of are attached to the vertex of the positive puncture of and oriented away from it. Then the boundary strata of are in one to one correspondence with such graphs and the components of the several level curve are in one-to-one correspondence with downwards oriented sub-trees consisting of one vertex and all edges emanating from it. For example the graph of a curve lying in the interior of is simply a vertex with edges attached and oriented away from the vertex. To distinguish the edges of such graphs , we call an edge a gluing edge if it is attached to two vertices and free if it is attached only to one vertex.

Note next that the induced asymptotic markers are compatible with the level structure in the boundary of in the sense that they vary continuously with the domain inside the compactification. To see this, note that in a boundary stratum corresponding to a graph , it is sufficient to study neck stretching for cylinders corresponding to linear subgraphs of , and here the compatibility of asymptotic markers with the level structure is obvious.

Consider the bundle , with , , of disks or spheres with punctures with cylindrical ends compatible with the markers. The fiber of this bundle is contractible so there exists a section. We next show that there is also a section over the compactification of . The proof is inductive. We first choose cylindrical ends for disks and spheres with 3 punctures. Gluing these we get cylindrical ends in a neighborhood of the boundary of the moduli space of disks and spheres with 4 punctures. Since the fiber of is contractible this choice can be extended continuously over the whole space of disks and spheres with 4 punctures. The proof then continues inductively in the obvious way: a choice of cylindrical ends for disks and spheres whose number of negative punctures is strictly smaller than determines a section of the bundle near its boundary, and this section extends because the fiber of the bundle is contractible.

Let , denote a system of sections as in the inductive construction above, with defined over the compactification of . We say that

 D=⋃h∈{0,1};m,k≥0Dh;hm,k

is a system of cylindrical ends that is compatible with breaking.

We identify with its graph and think of it as a subset of . The projection of onto is a homeomorphism and, after using smooth approximation, a diffeomorphism with respect to the natural stratification of the space determined by several level curves. Via this projection we endow with the structure of a set consisting of (several level) curves with additional data corresponding to a choice of a cylindrical end neighborhood at each puncture.

A neighborhood of a several level curve can then be described as follows. Consider the graph determined by . Let denote the vertices of with the top vertex and let denote the gluing edges of . Let be neighborhoods in of the component corresponding to . Then a neighborhood of is given by

 (2.1)

for . Here the gluing parameters measure the length of the breaking cylinder or strip corresponding to the gluing edge . More precisely, assume that connects and and corresponds to the curve of attached at its positive puncture to a negative puncture of the curve of . Then, given the cylindrical ends (interior case) or (boundary case) for , respectively (interior case) or (boundary case) for , the glued curve corresponding to the parameter is obtained via the gluing operation on these cylindrical ends defined by cutting out or from the cylindrical end of , cutting out or from the cylindrical end of , and gluing the remaining compact domains in the cylindrical ends by identifying with , respectively with . We refer to the resulting compact domain as the breaking cylinder or strip, and we refer to or as its middle circle or segment. Given a several level curve in this neighborhood we write for the closures of the components that remain if the middle circle or segment in each breaking cylinder or strip is removed, and that correspond to subsets of the levels of the broken curve. See Figure 3.

### 2.2. Almost complex structures

We next introduce splitting compatible families of almost complex structures over . Let denote the space of almost complex structures on compatible with and adapted to the contact form in the cylindrical end, i.e. if then in the cylindrical end preserves the contact planes and takes the vertical direction to the Reeb direction. Our construction of a family of almost complex structures is inductive. We start with strips, cylinders and cylinders with slits with coordinates . Here we require that depends only on the or coordinate. Assume that we have defined a family of almost complex structures for all curves , which have the form above in every cylindrical end and which commute with restriction to components for several level curves. By gluing we then have a field of almost complex structures in a neighborhood of the boundary of for . Since is contractible it is easy to see that we can extend this family to all of . We call the resulting family of almost complex structures over the universal curve corresponding to , splitting compatible.

### 2.3. A simplex bundle

Consider the trivial bundle

 Ehm+k−1=Dh;hm,k×Δhm+k−1→Dh;hm,k

over , with fiber the open -simplex

 Δhm+k−1={(s1,…,shm+k):∑isi=1, si>0}.

Since the bundle is trivial, it extends as such over the compactification of . We think of the coordinates of a point over a disk or sphere as representing weights at its negative punctures, and we think of the positive puncture as carrying the weight .

We next define restriction maps for over the boundary of . Let denote the weights of a several level curve in the boundary of with graph . Let be a component of this building corresponding to the vertex of , with positive puncture and negative punctures . Define the weight at , as follows. For , equals the sum of all weights at negative punctures of the total several level curve for which there exists a level-increasing path in from to . For , if the edge of the negative puncture is free then equals the weight of the puncture as a puncture of the total several level curve, and if the edge is a gluing edge connecting and , then equals the sum of all weights at negative punctures of the total several level curve for which there exists a level-increasing path in from to . Note that by construction.

The component restriction map then takes the point over to the point

 rj(s)=1w(q0)(w(q1),…,w(qn))∈Δn−1

over in . The component restriction map is defined on the restriction of to the stratum that corresponds to in the boundary of .

### 2.4. Superharmonic functions and non-positive 1 forms

Our main Floer homological constructions involve studying Floer holomorphic curves parameterized by finite dimensional families of 1-forms with values in Hamiltonian vector fields. As discussed in Section 1, it is important that the -forms are non-positive, i.e. the associated 2-forms are non-positive multiples of the area form. Furthermore, in order to derive basic homological algebra equations, the 1-forms must be gluing/breaking compatible on the boundary of Deligne-Mumford space. In this section we construct a family of superharmonic functions parameterized by that is compatible with the component restriction maps at several level curves. The differentials of these functions multiplied by the complex unit then give a family of 1-forms with non-positive exterior derivative that constitutes the basis for our construction of 0-order term in the Floer equation.

Fix a smooth decreasing function such that and

 (2.2) lims→0+κ(s)=+∞.

We will refer to as a stretching profile.

We will construct a family of functions over curves in parameterized by the bundle in the following sense. If belongs to the fiber over a one level curve then . If is a several level curve with graph and components corresponding to its vertices , , then is the collection of functions on , where denotes the component restriction map to . Our construction uses induction on the number of negative punctures and on the number of levels.

In the first case and the domain is the strip , the cylinder , or the cylinder with a slit (which we view as a subset of ). Over these domains the fiber of is a point and we take the function to be the projection to the -factor.

For , we specify properties of the functions separately for one level curves in the interior of and for a neighborhood of several level curves near the boundary. We start with one level curves. Let be a section of over one level curves in the interior . Let and write .

We say that a smooth family of functions over the interior satisfies the one level conditions if the following hold. (We write for the projection.)

• There is a constant such that in a neighborhood of infinity in the cylindrical end at the positive puncture

 (2.3) ge(s+it)=c0+s,

where is the complex coordinate in the cylindrical end, i.e. in for an interior puncture and in for a boundary puncture, see Section 2.1.

• There are constants , , , and for , such that in a neighborhood of infinity in the cylindrical end of the negative puncture of the form for interior punctures or for boundary punctures, we have , where

 (2.4) ge(s)={c′j+σwjs for −R≥s≥−R−κ(wj),cj+s for −R−κ(wj)−1≥s>−∞,

is a concave function, , and where is the stretching profile (2.2). In particular, for each weight at a negative puncture there is a cylinder or strip region of length at least along which , with .

• The function is superharmonic, everywhere.

• The derivative of in the direction of the normal of the boundary vanishes everywhere:

 ∂ge∂ν=0 along ∂Dh;hm,k.
###### Remark 2.1.

For the boundary condition , note that, for the cylinder with a slit, in local coordinates , at the end of the slit the standard function looks like , and .

###### Remark 2.2.

The appearance of the “extra factor” in (2.4) is to allow for a certain interpolation below. As we shall see, we can take arbitrarily close to on compact sets of . As mentioned in Section 1, one of the main uses of weights is to force solutions to degenerate for small weights, and for desired degenerations it is enough that be uniformly bounded. At the opposite end we find the following restriction on : superharmonicity in the cylindrical end near a negative puncture where the weight is implies that , and in particular if . In general, superharmonicity of the function is equivalent to the differential being non-positive with respect to the conformal area form on the domain . This is compatible with Stokes’ theorem, which gives

 ∫Dh;hm,k−d(i∗dge)=1−(hm+k)≤0.

We will next construct families of functions satisfying the one-level condition over any compact subset of the interior of . Later we will cover all of with a system of neighborhoods of the boundary where condition above is somewhat weakened but still strong enough to ensure degeneration for small weights.

###### Lemma 2.3.

If is a constant section, then, over any compact subset , there is a family of functions that satisfies the one level conditions. Moreover, we can take in arbitrarily close to .

###### Proof.

For simpler notation, let . Consider first the case when the positive puncture and all the negative punctures are interior. Fix an additional marked point in the domain. For each , fix a conformal map to which takes the positive puncture to , the marked point to some point in , and the negative puncture to . Fix and let be the function with the -coordinate on . Let be a concave approximation of this function with second derivative non-zero only on two intervals of finite length located near , linear of slope near and linear of slope near , see Figure 5. Consider the function

 g=k∑j=1gj.

Then is superharmonic but it does not quite have the right behavior at the punctures. Here however, the leading terms are correct and the errors are exponentially small. We turn off the exponential error in a neighborhood of in the region of support of the second derivative of . We can arrange the parameters so that the resulting function satisfies (2.3) near the positive puncture, and it satisfies the top equation in the right hand side of (2.4) in some neighborhood of . In order to achieve the bottom equation in a neighborhood of we simply replace the linear function of slope by a concave function that interpolates between it and the linear function of slope . The fact that we can take arbitrarily close to follows from the construction.

The case of boundary punctures can be treated in exactly the same way. In case of a positive boundary puncture and a negative interior puncture we replace the cylinder above with the cylinder with a slit along and in case of both positive and negative boundary punctures we use the cylinder with a slit all along . ∎

###### Remark 2.4.

For future reference we call the regions in the cylindrical ends where regions of concavity.

We next want to define a corresponding notion for several level curves. To this end we consider nested neighborhoods

 ⋯⊂Nℓ⊂Nℓ−1⊂Nℓ−2⊂⋯⊂N2,

were is a neighborhood of the subset of -level curves. Consider constant sections of over and let be a family of functions. The -level conditions are the same as the one level conditions , , and , and also the following new condition:

• For curves in with and any , there is a strip or cylinder region of length at least where for .

Our next lemma shows that there is a family of functions that satisfies the -level condition and that is also compatible with splittings into several level curves in the following sense.

We say that a family of functions as above is splitting compatible if the following holds. If , is a family of curves that converges as to an -level curve with components and if is any compact subset that converges to a compact subset of , then there is a sequence of constants such that the restriction converges to , where is the component restriction of to .

###### Lemma 2.5.

There exists a system of neighborhoods

 ⋯⊂Nℓ⊂Nℓ−1⊂Nℓ−2⊂⋯⊂N2,

and a splitting compatible family of functions parameterized by constant sections of that satisfies the -level condition for all .

###### Proof.

The proof is inductive. In the first case there are only one level curves and we use the canonical functions discussed above. Consider next a gluing compatible section over with . This space is an interval and the boundary points correspond to two level curves with both levels and in , . Consider a neighborhood of such a two level curve in parameterized by a gluing parameter , see (2.1). Assume that the positive puncture of is attached at a negative puncture of . Write , for the resulting domain and write for the part of the curve that is naturally a subset of , see the discussion in Section 2.1. Let and denote the functions of the component restrictions of to and . Then there is a constant such that

 (2.5) c(ρ)=gr0(e)|∂S0(ρ)−gr1(e)|∂S1(ρ).

We then define the function as

 ge(ρ)={gr0(e) on S0(ρ),c(ρ)+gr1(e) on S1(ρ).

Then is smooth and satisfies , , and and has the required properties for restrictions to levels. Furthermore, the restriction of to satisfies (2.4) with replaced by , where is the weight of at the negative puncture of where is attached (except that the interval in the second equation is not infinite but finite) and the restriction of to satisfies (2.4) with the weights of at the negative ends of . Let denote the weights at the negative punctures of which are negative punctures of , seen as negative punctures of . Then by definition

 wj=w(q0)wj(r1(e)).

Since

 (wj)12=(w(q0)wj(r1(e)))12≥min(w(q0),wj(r1(e)))

and

 κ((wj)12)≤κ(min(w(q0),wj(r1(e)))=max(κ(w(q0)),κ(wj(r1(e)))),

we find that there exists a strip or cylinder region of length at least where , with . Thus the two level condition holds.

We next want to extend the family of functions over all of , , respecting condition . To this end we consider a neighborhood of the broken curves in the boundary where the glued functions described above are defined. Using the gluing parameter this neighborhood can be identified with a half infinite interval. As the gluing parameter decreases we deform the derivative of the function as follows: we decrease it uniformly below the gluing region and stretch the region near the negative puncture where it is small, until we reach the level one function. See Figure 6. For this family conditions , , , and hold everywhere and holds in the compact subset of which is the complement of a suitable subset .

For more general two level curves with lying in we argue in exactly the same way using the gluing parameter to interpolate between the natural gluing of the functions of the component restrictions of and the function of , see Lemma 2.3, satisfying the one-level condition.

Consider next the general case. Assume that we have found a family of functions , associated to a constant section defined over the subset consisting of all curves in with at most levels, that satisfies conditions , , and everywhere, and assume that there are nested neighborhoods

 Nℓ⊂Nℓ−1⊂Nℓ−2⊂⋯⊂N2,

were is a neighborhood of in such that condition holds in .

Consider a curve in the boundary of with levels. Assume that the top level curve of has negative punctures at which there are curves of levels attached. Let denote the component restriction , . Our inductive assumption gives a smooth family of superharmonic functions with properties , , and for curves in a neighborhood of these broken configurations depending smoothly on . Denote the corresponding functions . Consider now a coordinate neighborhood of the form (2.1) around :

 U=U0×r∏i=1(0,∞)j×Uj.

Let . For curves , write for the curve that results from gluing these according to and in analogy with the two level case, write for the part of that is naturally a subset of . Our inductive assumption then shows that there are constants , , such that

 (2.6) cj(r0(e),rj(e),ρj)=g0r0(e)|∂jS0(ρ)−grj(e)|∂Sj(ρ),

where is the boundary component of where is attached. Define the function as

 ge(ρ)={gr0(e) on S0(ρ),grj(e)+cj(r0(e),rj(e),ρj) on Sj(ρ),j=1,…,r.

It is immediate that the function satisfies , , and . We show that condition holds. Let be a negative puncture in some , . Let

 wj′q=w